A Coarea–Fredholm Reduction Framework for a Multidimensional Cauchy-Type Integral Equation

1. Introduction

Multidimensional integral equations with kernels depending on scalar functions of the variables arise in reduction procedures for partial differential equations, potential theory, and operator-theoretic models of singular interactions. The model considered here is the Cauchy-type equation

$$\lambda u\left(x\right)-p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy=f\left(x\right),x\in \Omega ,$$

(1)

where $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, $\lambda \in \mathbb{C}$, and $\varphi ,\psi :\Omega \to \mathbb{R}$ are prescribed scalar functions. This problem is motivated by Maz’ya’s formulation of integral equations connected with the coarea formula and by the list of open problems concerning multidimensional equations whose singularities are governed by lower-dimensional variables [1,2]. The difficulty is that (1) is genuinely multidimensional, whereas the denominator has the one-dimensional Cauchy structure $\varphi \left(x\right)-\psi \left(y\right)$.

The coarea formula is the natural tool for separating these two features. In geometric measure theory, coarea identities express integration over $\Omega$ through integration over level sets and a scalar push-forward density [3,4]. In the present problem, this allows the integration in y to be organized according to the values of $\psi \left(y\right)$. The corresponding singular integral in the reduced variable is then a finite Cauchy transform, whose boundedness on $L^p$, $1<p<\infty$, follows from the classical Hilbert transform theory [5]. The resulting Fredholm analysis combines compact-operator theory [6,7] with the one-dimensional singular-integral theory developed in the classical literature [8,9,10,11,12].

Recent work also confirms that both components of this approach remain active. Modern coarea formulae and coarea-type inequalities have been developed in non-Euclidean and sub-Riemannian settings [16,17], while the finite Hilbert transform continues to be investigated through optimal-domain, spectral, Zygmund-space, and measure-theoretic methods [13,14,15,18,19]. These developments are not used as black-box inputs in the proofs below, but they show that the coarea and finite-Hilbert-transform mechanisms isolated here are part of a currently active analytical framework. Recent numerical work on Cauchy singular integral equations also emphasizes the continuing relevance of finite-Hilbert-transform models beyond purely abstract Fredholm theory [20].

The purpose of this paper is deliberately limited. We do not claim to solve the fully general degenerate form of Maz’ya’s problem. Instead, we establish a rigorous coarea–Fredholm framework under explicit geometric and mapping hypotheses. The central factorization is

$$K=A_{\varphi }{B}_{\psi },$$

(2)

where K is the multidimensional Cauchy-type operator in (1), $A_{\varphi }h=h\circ \varphi$ is a lifting operator, and $B_{\psi }$ is a finite Cauchy transform applied to the $\psi$-level-set coarea trace of the unknown. For $\lambda \ne 0$, this factorization converts (1) into the reduced equation

$$\lambda h-Sh=B_{\psi }f,S=B_{\psi }{A}_{\varphi },$$

(3)

with reconstruction

$$u\left(x\right)={\displaystyle \frac{f\left(x\right)+h\left(\varphi \right(x\left)\right)}{\lambda }}.$$

(4)

Thus the analytical content of the multidimensional equation is transferred to an operator acting on functions of one real variable.

The paper separates three regimes. First, if the essential ranges of $\varphi$ and $\psi$ are separated, the Cauchy kernel is regular after reduction and the reduced operator is compact. The standard Riesz–Schauder Fredholm alternative then gives solvability up to finitely many compatibility conditions. Second, if the ranges overlap but the map $F=(\varphi ,\psi )$ has rank two on the relevant interaction region, the singularity is transversal. Under Hölder regularity of the associated two-variable coarea density, level-set coarea integration produces compact regularization before the finite Cauchy transform is applied. Third, if $\psi =\varphi$, the singularity is not removed by coarea averaging. The reduced equation becomes a weighted finite Hilbert transform equation, and the appropriate Fredholm framework is the classical Noether theory of one-dimensional singular integral operators [8,9,10].

Several technical restrictions are essential. The lifting and level-set coarea integration operators are not automatically bounded on unweighted $L^p$ spaces; therefore, we impose explicit boundedness assumptions on the coarea densities $a_{\varphi }$ and $a_{\psi }$. Similarly, the transversal case requires more than the formal statement that level sets intersect regularly: it requires a rank-two condition for $(\varphi ,\psi )$ and regularity of the two-variable density. These hypotheses are strong, but they make the reduction theorem and the Fredholm conclusions transparent and verifiable.

The contribution of the paper is therefore threefold: (i) an exact operator factorization and reduction theorem for $\lambda \ne 0$; (ii) compact Fredholm solvability for separated-range and transversal-overlap geometries; and (iii) an exact reduction of the diagonal case to a weighted finite Hilbert transform, clarifying why this case belongs to the non-compact singular-integral regime. Section 2 fixes the functional setting and proves the boundedness of the coarea lifting and averaging operators. Section 3 proves the factorization and the exact reduction theorem. Section 4 gives the Fredholm analysis in the separated, transversal, and diagonal cases. Section 5 presents model examples. Section 6 and Section 7 discuss the scope, limitations, and conclusions of the work.

2. Materials and Methods: Functional Setting and Coarea Operators

Let $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, be a bounded measurable domain. Throughout the paper,

$$1<p<\infty ,p^{\prime }={\displaystyle \frac{p}{p-1}}.$$

Unless otherwise stated, $\varphi ,\psi \in C^1\left(\overline{\Omega }\right)$. In the transversal part we assume $C^2$-regularity. Let I and J be bounded intervals containing the essential ranges of $\varphi$ and $\psi$, respectively. Functions on I or J are extended by zero outside their intervals when convenient.

For a scalar function $\chi :\Omega \to \mathbb{R}$, define its coarea density by

$$a_{\chi }\left(s\right)={\int }_{{\chi }^{-1}\left(s\right)}{\displaystyle \frac{1}{|\nabla \chi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right)$$

(5)

for almost every regular value s, whenever the push-forward measure of Lebesgue measure under $\chi$ is absolutely continuous. Equivalently,

$${\chi }_{\#}\left(dy\right)=a_{\chi }\left(s\right)ds.$$

The coarea formula gives

$${\int }_{\Omega }g\left(y\right)dy={\int }_{\mathbb{R}}{\int }_{{\chi }^{-1}\left(s\right)}{\displaystyle \frac{g\left(y\right)}{|\nabla \chi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right)ds$$

(6)

for nonnegative measurable g, and hence for integrable g by the usual decomposition into positive and negative parts.

Definition 2.1 

(Lifting and level-set coarea trace). For $h:I\to \mathbb{C}$, define

$$\left(A_{\varphi }h\right)\left(x\right)=h\left(\varphi \left(x\right)\right),x\in \Omega .$$

(7)

For $u:\Omega \to \mathbb{C}$, define the ψ-level-set coarea trace

$$\left({\Pi }_{\psi }u\right)\left(r\right)={\int }_{{\psi }^{-1}\left(r\right)}{\displaystyle \frac{u\left(y\right)}{|\nabla \psi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right),r\in J,$$

(8)

whenever the integral is meaningful. This is an unnormalized coarea trace, not a normalized average. A normalized average would require division by the density $a_{\psi }\left(r\right)$ on the set where $a_{\psi }\left(r\right)>0$; the unnormalized form is the one that appears directly in the coarea reduction of the integral equation.

Proposition 2.1 

(Boundedness of lifting). Assume $a_{\varphi }\in L^{\infty }\left(I\right)$. Then

$$A_{\varphi }:L^p\left(I\right)⟶L^p(\Omega )$$

is bounded and

$$\parallel A_{\varphi }{h\parallel }_{L^p(\Omega )}\le \parallel a_{\varphi }{\parallel }_{L^{\infty }\left(I\right)}^{1/p}{\parallel h\parallel }_{L^p\left(I\right)}.$$

(9)

More precisely,

$$\parallel A_{\varphi }{h\parallel }_{L^p(\Omega )}^p={\int }_I{|h\left(s\right)|}^p{a}_{\varphi }\left(s\right)ds.$$

(10)

Proof. 

Applying (6) to $g\left(x\right)={|h\left(\varphi \left(x\right)\right)|}^p$ gives

$${\int }_{\Omega }{|h\left(\varphi \left(x\right)\right)|}^{p}dx={\int }_I{|h\left(s\right)|}^p{\int }_{{\varphi }^{-1}\left(s\right)}{\displaystyle \frac{1}{|\nabla \varphi (x)|}}d{\mathcal{H}}^{n-1}\left(x\right)ds.$$

This proves (10); (9) follows immediately. □

Proposition 2.2 

(Boundedness of level-set coarea integration). Assume $a_{\psi }\in L^{\infty }\left(J\right)$. Then

$${\Pi }_{\psi }:L^p(\Omega )⟶L^p\left(J\right)$$

is bounded and

$$\parallel {\Pi }_{\psi }{u\parallel }_{L^p\left(J\right)}\le \parallel a_{\psi }{\parallel }_{L^{\infty }\left(J\right)}^{1/p^{\prime }}{\parallel u\parallel }_{L^p(\Omega )}.$$

(11)

Proof. 

For almost every $r\in J$, Hölder’s inequality on the level set ${\psi }^{-1}\left(r\right)$ gives

$$|{\Pi }_{\psi }{u\left(r\right)|}^p\le {\left({\int }_{{\psi }^{-1}\left(r\right)}{\displaystyle \frac{1}{|\nabla \psi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right)\right)}^{p-1}{\int }_{{\psi }^{-1}\left(r\right)}{\displaystyle \frac{{|u\left(y\right)|}^p}{|\nabla \psi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right).$$

Therefore,

$$|{\Pi }_{\psi }{u\left(r\right)|}^p\le a_{\psi }{\left(r\right)}^{p-1}{\int }_{{\psi }^{-1}\left(r\right)}{\displaystyle \frac{{|u\left(y\right)|}^p}{|\nabla \psi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right).$$

Integrating in r and using the coarea formula gives

$$\parallel {\Pi }_{\psi }{u\parallel }_{L^p\left(J\right)}^p\le \parallel a_{\psi }{\parallel }_{L^{\infty }\left(J\right)}^{p-1}{\parallel u\parallel }_{L^p(\Omega )}^p.$$

Taking the p-th root proves (11). □

For bounded intervals $I,J\subset \mathbb{R}$, define the finite Cauchy transform from J to I by

$$\left(C_J^{I}v\right)\left(t\right)=p.v.{\int }_J{\displaystyle \frac{v\left(r\right)}{t-r}}dr,t\in I.$$

(12)

When $t\notin J$, this is an ordinary integral. When $t\in J$, it is understood in the principal-value sense. This is the finite-interval version of the classical Cauchy singular integral, closely related to the Hilbert transform on the line [5,8].

Throughout the paper, the multidimensional principal value is understood through the same coarea truncation. More precisely, for fixed x we write

$$p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy=\underset{\epsilon \downarrow 0}{lim}{\int }_{\{y\in \Omega :|\varphi \left(x\right)-\psi \left(y\right)|>\epsilon \}}{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy,$$

(13)

whenever this limit exists. In the $L^p$-operator formulation used below, this principal value is defined equivalently by the coarea identity

$$K=A_{\varphi }{C}_J^I{\Pi }_{\psi }.$$

This convention avoids treating the multidimensional singularity as an unspecified pointwise object; the singular variable is the one-dimensional coarea variable $r=\psi \left(y\right)$. We do not assert that this coarea principal value is automatically identical to every possible multidimensional principal-value prescription. Equivalence with other truncation procedures must be checked separately in any application.

Lemma 2.1 

(Boundedness of the finite Cauchy transform). Let $I,J\subset \mathbb{R}$ be bounded intervals and $1<p<\infty$. Then

$$C_J^I:L^p\left(J\right)⟶L^p\left(I\right)$$

is bounded.

Proof. 

Extend $v\in L^p\left(J\right)$ by zero to $\tilde{v}\in L^p\left(\mathbb{R}\right)$. Up to the normalization convention, $C_J^{I}v$ is the restriction to I of the Hilbert transform of $\tilde{v}$. Since the Hilbert transform is bounded on $L^p\left(\mathbb{R}\right)$ for $1<p<\infty$,

$$\parallel C_J^I{v\parallel }_{L^p\left(I\right)}\le C_p\parallel \tilde{v}{\parallel }_{L^p\left(\mathbb{R}\right)}=C_p{\parallel v\parallel }_{L^p\left(J\right)}.$$

□

Combining Proposition 2.2 and Lemma 2.1, define

$$B_{\psi }=C_J^I{\Pi }_{\psi }:L^p(\Omega )⟶L^p\left(I\right).$$

(14)

This operator is bounded whenever $a_{\psi }\in L^{\infty }\left(J\right)$.

3. Results I: Coarea Factorization and Exact Reduction

Let

$$X=L^p(\Omega ),Y=L^p\left(I\right).$$

Assume in this section that

$$a_{\varphi }\in L^{\infty }\left(I\right),a_{\psi }\in L^{\infty }\left(J\right).$$

(15)

Then $A_{\varphi }:Y\to X$ and $B_{\psi }:X\to Y$ are bounded.

Define the multidimensional operator by

$$\left(Ku\right)\left(x\right)=\left(A_{\varphi }{B}_{\psi }u\right)\left(x\right)=\left(B_{\psi }u\right)\left(\varphi \left(x\right)\right).$$

(16)

Equivalently, for functions for which the principal value is meaningful pointwise,

$$\left(Ku\right)\left(x\right)=p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy.$$

(17)

Formula (16) is the natural $L^p$-operator interpretation of (17).

Proposition 3.1 

(Coarea factorization). Under (15), the operator $K:X\to X$ is bounded and satisfies

$$K=A_{\varphi }{B}_{\psi }.$$

(18)

Moreover, for functions for which the principal value can be evaluated through coarea decomposition,

$$p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy=\left(C_J^I{\Pi }_{\psi }u\right)\left(\varphi \left(x\right)\right).$$

(19)

Proof. 

Boundedness follows from

$${\parallel Ku\parallel }_X=\parallel A_{\varphi }{B}_{\psi }{u\parallel }_X\le \parallel A_{\varphi }{\parallel }_{Y\to X}\parallel B_{\psi }{\parallel }_{X\to Y}{\parallel u\parallel }_X.$$

For smooth functions for which the manipulations are justified, the coarea formula gives

$$p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy=p.v.{\int }_J{\displaystyle \frac{1}{\varphi \left(x\right)-r}}{\int }_{{\psi }^{-1}\left(r\right)}{\displaystyle \frac{u\left(y\right)}{|\nabla \psi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right)dr.$$

The right-hand side is exactly $\left(C_J^I{\Pi }_{\psi }u\right)\left(\varphi \left(x\right)\right)$. The general $L^p$ formulation is obtained by the bounded operator definition (16). □

Define the reduced operator

$$S=B_{\psi }{A}_{\varphi }:Y⟶Y.$$

(20)

Formally,

$$\left(Sh\right)\left(t\right)=p.v.{\int }_{\Omega }{\displaystyle \frac{h\left(\varphi \right(y\left)\right)}{t-\psi \left(y\right)}}dy.$$

(21)

Lemma 3.1 

(Fredholm transfer for the products $AB$ and $BA$). Let X and Y be Banach spaces, let $A:Y\to X$ and $B:X\to Y$ be bounded operators, and let $\lambda \ne 0$. Then

$$\lambda I_X-AB\mathrm{is}\mathrm{Fredholm}\mathrm{on}X$$

if and only if

$$\lambda I_Y-BA\mathrm{is}\mathrm{Fredholm}\mathrm{on}Y.$$

In that case the two Fredholm indices are equal. Moreover, B gives an isomorphism

$$ker(\lambda I_X-AB)\simeq ker(\lambda I_Y-BA),$$

with inverse ${\lambda }^{-1}A$, and the adjoint correspondence gives the analogous identification of compatibility conditions.

Proof. 

This is the standard nonzero spectral and Fredholm transfer for the operator products $AB$ and $BA$; it follows, for example, from the usual product theorem for Fredholm operators applied to $I-{\lambda }^{-1}AB$ and $I-{\lambda }^{-1}BA$[6,7]. The explicit kernel correspondence is immediate: if $ABx=\lambda x$, then $Bx\ne 0$ unless $x=0$, and $BA\left(Bx\right)=\lambda Bx$. Conversely, if $BAy=\lambda y$, then $Ay\ne 0$ unless $y=0$, and $AB\left(Ay\right)=\lambda Ay$. The inverse maps are B and ${\lambda }^{-1}A$. Applying the same argument to the Banach adjoints gives the cokernel correspondence and hence the compatibility conditions. □

Theorem 3.1 

(Exact coarea reduction). Assume (15) and let $\lambda \ne 0$. For $f\in X$, a function $u\in X$ solves

$$\lambda u-Ku=f$$

(22)

if and only if $h=B_{\psi }u\in Y$ solves

$$\lambda h-Sh=B_{\psi }f.$$

(23)

In that case the solution is reconstructed by

$$u={\displaystyle \frac{f+A_{\varphi }h}{\lambda }},\mathrm{i}.\mathrm{e}.u\left(x\right)={\displaystyle \frac{f\left(x\right)+h\left(\varphi \right(x\left)\right)}{\lambda }}.$$

(24)

Proof. 

Assume that u solves (22). Since $K=A_{\varphi }{B}_{\psi }$, set $h=B_{\psi }u$. Then

$$\lambda u-A_{\varphi }h=f,\mathrm{so}u={\displaystyle \frac{f+A_{\varphi }h}{\lambda }}.$$

Applying $B_{\psi }$ to this identity gives

$$h={\displaystyle \frac{B_{\psi }f+B_{\psi }{A}_{\varphi }h}{\lambda }}={\displaystyle \frac{B_{\psi }f+Sh}{\lambda }},$$

which is (23).

Conversely, suppose that $h\in Y$ solves (23) and define u by (24). Then

$$B_{\psi }u={\displaystyle \frac{B_{\psi }f+B_{\psi }{A}_{\varphi }h}{\lambda }}={\displaystyle \frac{B_{\psi }f+Sh}{\lambda }}=h,$$

where the last equality is (23). Hence $Ku=A_{\varphi }{B}_{\psi }u=A_{\varphi }h$, and therefore

$$\lambda u-Ku=f+A_{\varphi }h-A_{\varphi }h=f.$$

□

Corollary 3.1 

(Fredholm equivalence of the full and reduced equations). Under the assumptions of Theorem 3.1, let $K=A_{\varphi }{B}_{\psi }$ on $X=L^p(\Omega )$ and $S=B_{\psi }{A}_{\varphi }$ on $Y=L^p\left(I\right)$. For every $\lambda \ne 0$,

$$\lambda I_X-K\mathrm{is}\mathrm{Fredholm}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\lambda I_Y-S\mathrm{is}\mathrm{Fredholm}.$$

The indices are equal, the nonzero eigenvalue obstructions correspond through $B_{\psi }$ and $A_{\varphi }$, and the Fredholm compatibility conditions for the multidimensional equation are precisely the compatibility conditions for the reduced equation

$$(\lambda I_Y-S)h=B_{\psi }f.$$

Proof. 

This is Lemma 3.1 with $A=A_{\varphi }$ and $B=B_{\psi }$, together with Theorem 3.1. □

Remark 3.1 

(The case $\lambda =0$). The reduction theorem is stated for $\lambda \ne 0$, which is the natural regime for Fredholm alternatives of the form $\lambda I-S$. The case $\lambda =0$ becomes the range equation $Ku=-f$, equivalently $A_{\varphi }{B}_{\psi }u=-f$, and requires a separate analysis of the range of the product operator.

4. Results II: Fredholm Regimes

4.1. Separated Ranges

Assume

$$1.7em(I,J)>0.$$

(25)

Then the kernel ${(t-r)}^{-1}$ is continuous on $\overline{I}\times \overline{J}$, and no principal value is involved. Define

$$\left(Tv\right)\left(t\right)={\int }_J{\displaystyle \frac{v\left(r\right)}{t-r}}dr,t\in I.$$

(26)

Then $B_{\psi }=T{\Pi }_{\psi }$.

Lemma 4.1 

(Compactness of the regular kernel). Under (25), the operator $T:L^p\left(J\right)\to L^p\left(I\right)$ defined by (26) is compact.

Proof. 

The kernel $k(t,r)={(t-r)}^{-1}$ is continuous on $\overline{I}\times \overline{J}$. For ${\parallel v\parallel }_{L^p\left(J\right)}\le 1$, Hölder’s inequality gives a uniform bound on $Tv$. Moreover,

$$|\left(Tv\right)\left(t_1\right)-\left(Tv\right)\left(t_2\right)|\le \parallel k(t_1,·)-k(t_2,·){\parallel }_{L^{p^{\prime }}\left(J\right)}{\parallel v\parallel }_{L^p\left(J\right)},$$

and the right-hand side tends to zero uniformly as $t_1\to t_2$, because k is uniformly continuous. Hence T maps the unit ball of $L^p\left(J\right)$ into a bounded equicontinuous subset of $C\left(\overline{I}\right)$. By the Arzelà–Ascoli theorem and the continuous embedding $C\left(\overline{I}\right)\hookrightarrow L^p\left(I\right)$, T is compact. □

Theorem 4.1 

(Fredholm alternative in the separated case). Let $1<p<\infty$. Assume

$$1.7em(I,J)>0,a_{\varphi }\in L^{\infty }\left(I\right),a_{\psi }\in L^{\infty }\left(J\right).$$

Then $S=B_{\psi }{A}_{\varphi }:L^p\left(I\right)\to L^p\left(I\right)$ is compact. Consequently, for every $\lambda \ne 0$, equation (1) satisfies the Fredholm alternative in the following reduced sense:

(i)

If $ker(\lambda I-S)=\left\{0\right\}$, then for every $f\in L^p(\Omega )$ there exists a unique solution $u\in L^p(\Omega )$, namely

$$u={\displaystyle \frac{f+A_{\varphi }h}{\lambda }},h={(\lambda I-S)}^{-1}{B}_{\psi }f.$$

(ii)

If $ker(\lambda I-S)\ne \left\{0\right\}$, then (1) is solvable if and only if

$$\ell \left(B_{\psi }f\right)=0\mathrm{for}\mathrm{all}\ell \in ker\left({(\lambda I-S)}^{*}\right).$$

(iii)

The exceptional values of λ in $\mathbb{C}\setminus \left\{0\right\}$ are isolated eigenvalues of S of finite algebraic multiplicity, with possible accumulation only at 0.

Proof. 

By Proposition 2.2, ${\Pi }_{\psi }:L^p(\Omega )\to L^p\left(J\right)$ is bounded. By Lemma 4.1, $T:L^p\left(J\right)\to L^p\left(I\right)$ is compact. Hence $B_{\psi }=T{\Pi }_{\psi }:L^p(\Omega )\to L^p\left(I\right)$ is compact. Since $A_{\varphi }:L^p\left(I\right)\to L^p(\Omega )$ is bounded by Proposition 2.1, the composition $S=B_{\psi }{A}_{\varphi }$ is compact on $L^p\left(I\right)$. The assertions follow from the Riesz–Schauder Fredholm alternative applied to $\lambda I-S$ [6,7], together with the reconstruction formula in Theorem 3.1. □

4.2. Transversal Singularities

We now consider the case $I\cap J\ne ⌀$, where the denominator in (1) may vanish. Let

$$F=(\varphi ,\psi ):\Omega \to {\mathbb{R}}^2.$$

In this subsection, assume $\varphi ,\psi \in C^2\left(\overline{\Omega }\right)$. We also assume that F is nondegenerate in the two-dimensional coarea sense on the interaction region used by the reduction:

$$rankDF\left(y\right)=2\mathrm{for}y\in F^{-1}(I\times J).$$

(27)

Equivalently,

$$J_F\left(y\right)=|\nabla \varphi \left(y\right)\wedge \nabla \psi \left(y\right)|>0\mathrm{on}{F}^{-1}(I\times J).$$

(28)

When I and J are chosen as the full essential ranges, this is a global transversality assumption on the relevant part of $\Omega$. The statement is local in the range variables: if critical or boundary values prevent (27) or the regularity condition below from holding on the full ranges, the same argument applies on compact subintervals $I_0⋐I$, $J_0⋐J$ for which the hypotheses are valid. The present formulation keeps a clean unweighted $L^p$ notation.

Define the two-variable coarea density

$$\rho (s,r)={\int }_{\{y\in \Omega :\varphi (y)=s,\psi (y)=r\}}{\displaystyle \frac{1}{J_F\left(y\right)}}d{\mathcal{H}}^{n-2}\left(y\right).$$

(29)

We impose the regularity condition

$$\rho \in C^{\alpha }(\overline{I}\times \overline{J}),0<\alpha <1.$$

(30)

This condition may fail near boundary values or critical values; in such cases one should work on interior subintervals or in weighted spaces adapted to the coarea density.

Lemma 4.2 

(Averaged composition). Assume (27). For $h\in L^p\left(I\right)$, define

$$\left(G_{\rho }h\right)\left(r\right)={\int }_I\rho (s,r)h\left(s\right)ds,r\in J.$$

(31)

Then, for almost every $r\in J$,

$${\Pi }_{\psi }\left(A_{\varphi }h\right)\left(r\right)=G_{\rho }h\left(r\right).$$

(32)

Consequently,

$$S=B_{\psi }{A}_{\varphi }=C_J^I{G}_{\rho }.$$

(33)

Proof. 

Fix a regular value r of $\psi$. On the hypersurface ${\psi }^{-1}\left(r\right)$, apply the coarea formula to the restriction of $\varphi$. The tangential Jacobian of this restriction is

$$J_{\varphi |{\psi }^{-1}\left(r\right)}\left(y\right)={\displaystyle \frac{|\nabla \varphi (y)\wedge \nabla \psi (y)|}{|\nabla \psi (y)|}}={\displaystyle \frac{J_F\left(y\right)}{|\nabla \psi (y)|}}.$$

Thus

$$\begin{array}{cc}\hfill {\Pi }_{\psi }\left(A_{\varphi }h\right)\left(r\right)& ={\int }_{{\psi }^{-1}\left(r\right)}{\displaystyle \frac{h\left(\varphi \right(y\left)\right)}{|\nabla \psi (y)|}}d{\mathcal{H}}^{n-1}\left(y\right)\hfill \\ \hfill & ={\int }_{I}h\left(s\right){\int }_{\{\varphi =s,\psi =r\}}{\displaystyle \frac{1}{J_F\left(y\right)}}d{\mathcal{H}}^{n-2}\left(y\right)ds\hfill \\ \hfill & ={\int }_I\rho (s,r)h\left(s\right)ds.\hfill \end{array}$$

Applying $C_J^I$ gives (33). □

Lemma 4.3 

(Compactness of transversal averaging). Assume (30). Then

$$G_{\rho }:L^p\left(I\right)⟶C^{\alpha }\left(\overline{J}\right)$$

is bounded. Consequently, as an operator from $L^p\left(I\right)$ to $L^p\left(J\right)$, $G_{\rho }$ is compact.

Proof. 

For $h\in L^p\left(I\right)$, Hölder’s inequality gives

$$|G_{\rho }{h\left(r\right)|\le \parallel \rho (·,r)\parallel }_{L^{p^{\prime }}\left(I\right)}{\parallel h\parallel }_{L^p\left(I\right)}\le C{\parallel h\parallel }_{L^p\left(I\right)}.$$

Similarly, for $r_1,r_2\in J$,

$$|G_{\rho }h\left(r_1\right)-G_{\rho }h\left(r_2\right)|\le \parallel \rho (·,r_1)-\rho (·,r_2){\parallel }_{L^{p^{\prime }}\left(I\right)}{\parallel h\parallel }_{L^p\left(I\right)}\le C|r_1-r_2{|}^{\alpha }{\parallel h\parallel }_{L^p\left(I\right)}.$$

Therefore bounded subsets of $L^p\left(I\right)$ are mapped into bounded subsets of $C^{\alpha }\left(\overline{J}\right)$. The embedding $C^{\alpha }\left(\overline{J}\right)\hookrightarrow L^p\left(J\right)$ is compact by Arzelà–Ascoli. Hence $G_{\rho }:L^p\left(I\right)\to L^p\left(J\right)$ is compact. □

Theorem 4.2 

(Fredholm solvability in the transversal singular case). Let $1<p<\infty$. Assume

$$a_{\varphi }\in L^{\infty }\left(I\right),a_{\psi }\in L^{\infty }\left(J\right),$$

and assume the transversal hypotheses (27) and (30). Then

$$S=B_{\psi }{A}_{\varphi }=C_J^I{G}_{\rho }:L^p\left(I\right)⟶L^p\left(I\right)$$

is compact. Consequently, for every $\lambda \ne 0$, the multidimensional equation (1), interpreted through the coarea principal value, satisfies the Fredholm alternative stated in Theorem 4.1.

Proof. 

By Lemma 4.3, $G_{\rho }:L^p\left(I\right)\to L^p\left(J\right)$ is compact. By Lemma 2.1, $C_J^I:L^p\left(J\right)\to L^p\left(I\right)$ is bounded. Therefore the composition $S=C_J^I{G}_{\rho }$ is compact on $L^p\left(I\right)$. The Fredholm alternative follows from the Riesz–Schauder theorem [6,7], and the equivalence with the original multidimensional equation follows from Theorem 3.1. □

Remark 4.1 

(Why transversality regularizes the singularity). The compactness in Theorem 4.2 does not come from the Cauchy transform itself, which is only bounded on $L^p$. It comes from the smoothing effect of the transversal level-set coarea integration $G_{\rho }$. The rank condition separates the variables $s=\varphi \left(y\right)$ and $r=\psi \left(y\right)$, and the Hölder regularity of ρ forces $G_{\rho }h$ to be Hölder continuous in r, even when $h\in L^p\left(I\right)$ only. The finite Cauchy transform is then applied after this compact regularization.

4.3. The Diagonal and Functionally Dependent Case

The transversal argument excludes the most singular situation in which $\varphi$ and $\psi$ are functionally dependent. The principal model is

$$\psi =\varphi .$$

(34)

Then

$$\left(Ku\right)\left(x\right)=p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\varphi \left(y\right)}}dy.$$

(35)

Assume $a_{\varphi }\in L^{\infty }\left(I\right)$. Then, for $h\in L^p\left(I\right)$,

$${\Pi }_{\varphi }\left(A_{\varphi }h\right)\left(s\right)=a_{\varphi }\left(s\right)h\left(s\right).$$

(36)

Therefore the reduced operator is

$$\left(H_{a_{\varphi }}h\right)\left(t\right)=p.v.{\int }_I{\displaystyle \frac{a_{\varphi }\left(s\right)h\left(s\right)}{t-s}}ds.$$

(37)

The reduced equation is

$$\lambda h\left(t\right)-p.v.{\int }_I{\displaystyle \frac{a_{\varphi }\left(s\right)h\left(s\right)}{t-s}}ds=B_{\varphi }f\left(t\right).$$

(38)

Theorem 4.3 

(Reduction to a weighted finite Hilbert transform). Assume $\psi =\varphi$, $a_{\varphi }\in L^{\infty }\left(I\right)$, and $1<p<\infty$. Then the multidimensional equation (1) is equivalent, for $\lambda \ne 0$, to the weighted finite Hilbert transform equation (38). Once (38) is solved for h, the solution of the multidimensional equation is reconstructed by

$$u\left(x\right)={\displaystyle \frac{f\left(x\right)+h\left(\varphi \right(x\left)\right)}{\lambda }}.$$

Proof. 

Formula (36) follows directly from the coarea formula. Hence

$$S=B_{\varphi }{A}_{\varphi }=H_{a_{\varphi }}.$$

The assertion follows from Theorem 3.1 with $\psi =\varphi$. □

The diagonal case is not compact in general. If $a_{\varphi }$ is sufficiently regular, then $\lambda I-H_{a_{\varphi }}$ belongs to the classical algebra of one-dimensional singular integral operators. With the normalization used in (12), the two interior boundary values of the local symbol are

$${\sigma }_{\pm }(s,\lambda )=\lambda \pm i\pi a_{\varphi }\left(s\right),s\in I.$$

(39)

Thus the interior ellipticity condition is

$$\lambda +i\pi a_{\varphi }\left(s\right)\ne 0\mathrm{and}\lambda -i\pi a_{\varphi }\left(s\right)\ne 0,s\in I.$$

(40)

Endpoint symbols and indices depend on the chosen scale of spaces, for example unweighted or weighted $L^p$-spaces. We therefore state the Fredholm transfer in a form that is precise and does not hide these classical endpoint conditions.

Corollary 4.1 

(Fredholm transfer in the diagonal case). Assume the hypotheses of Theorem 4.3. Let Y be a Banach function space on I such that

$$A_{\varphi }:Y\to L^p(\Omega ),B_{\varphi }:L^p(\Omega )\to Y,H_{a_{\varphi }}:Y\to Y$$

are bounded. Suppose that the one-dimensional operator

$$L_{\lambda }=\lambda I-H_{a_{\varphi }}$$

is Fredholm on Y and that $B_{\varphi }f\in Y$. Then the multidimensional equation (1) is Fredholm in the reduced sense: solvability is equivalent to the finite number of compatibility conditions for

$$L_{\lambda }h=B_{\varphi }f,$$

and all multidimensional solutions are given by

$$u={\displaystyle \frac{f+A_{\varphi }h}{\lambda }}.$$

In particular, if $L_{\lambda }$ is invertible on Y, then (1) has the unique solution corresponding to $h=L_{\lambda }^{-1}{B}_{\varphi }f$.

Remark 4.2 

(Role of one-dimensional Noether theory). Classical Noether theory for singular integral operators gives the Fredholm criterion for $L_{\lambda }$ in terms of interior and endpoint symbols and computes the index [8,9,10]. Modern work on the finite Hilbert transform further clarifies its spectra, optimal domains, and endpoint-sensitive function-space behavior [13,15,18,19]. The contribution here is not to replace that theory, but to show that the multidimensional diagonal problem is exactly reducible to it. This is qualitatively different from the separated and transversal cases, where compact Fredholm theory is sufficient.

4.4. Unified Solvability Principle

The preceding results can be summarized as follows.

Theorem 4.4 

(Coarea-Fredholm solvability principle). Let $1<p<\infty$, let $\Omega \subset {\mathbb{R}}^n$ be bounded, and let $I,J$ be bounded intervals containing the essential ranges of ϕ and ψ. Assume

$$a_{\varphi }\in L^{\infty }\left(I\right),a_{\psi }\in L^{\infty }\left(J\right).$$

Then the multidimensional equation (1), interpreted through the coarea principal value, is equivalent for $\lambda \ne 0$ to

$$\lambda h-Sh=B_{\psi }f,S=B_{\psi }{A}_{\varphi }.$$

Moreover:

(i)

If $(I,J)>0$, then S is compact on $L^p\left(I\right)$, and the Fredholm alternative applies.

(ii)

If $I\cap J\ne ⌀$, $F=(\varphi ,\psi )$ has rank two on $F^{-1}(I\times J)$, and the density ρ in (29) belongs to $C^{\alpha }(\overline{I}\times \overline{J})$, then $S=C_J^I{G}_{\rho }$ is compact on $L^p\left(I\right)$, and the Fredholm alternative applies.

(iii)

If $\psi =\varphi$, then $S=H_{a_{\varphi }}$, and the reduced problem is a weighted finite Hilbert transform equation. Fredholm solvability is governed by the classical one-dimensional singular-integral theory rather than by compactness.

In all cases, solutions of the multidimensional equation are reconstructed by

$$u\left(x\right)={\displaystyle \frac{f\left(x\right)+h\left(\varphi \right(x\left)\right)}{\lambda }}.$$

Proof. 

The equivalence and reconstruction formula follow from Theorem 3.1. Part (i) is Theorem 4.1; part (ii) is Theorem 4.2; part (iii) is Theorem 4.3 and Corollary 4.1. □

5. Examples

Example 5.1 

(A nonlinear separated-range model). Let $\Omega ={(0,1)}^2$,

$$\varphi (x_1,x_2)=x_1+2+\epsilon sin\left(2\pi x_2\right),\psi (y_1,y_2)=y_1,$$

where $0<\epsilon <1/4$. Then $\psi (\Omega )=(0,1)$, while $\varphi (\Omega )\subset (2-\epsilon ,3+\epsilon )$. Hence

$$1.7em\left(\varphi \right(\Omega ),\psi (\Omega \left)\right)\ge 1-\epsilon >0.$$

Moreover, $a_{\psi }\left(r\right)=1$. For $a_{\varphi }$, parameterizing level sets by $x_2$ gives

$$x_1=s-2-\epsilon sin\left(2\pi x_2\right),$$

whenever this value lies in $(0,1)$. Since

$$|\nabla \varphi |=\sqrt{1+4{\pi }^2{\epsilon }^2{cos}^2\left(2\pi x_2\right)}$$

and the arclength element along the graph is the same factor times $d{x}_2$, one obtains $a_{\varphi }\left(s\right)\le 1$. Thus the hypotheses of Theorem 4.1 hold, and the reduced equation is a compact Fredholm equation.

Example 5.2 

(A transversal singular model). Let $\Omega ={(0,1)}^2$,

$$\varphi (x_1,x_2)=x_1,\psi (x_1,x_2)=x_2.$$

Then we may take $I=J=[0,1]$ up to null endpoint changes, so the ranges overlap. However,

$$\nabla \varphi =(1,0),\nabla \psi =(0,1),J_F=|\nabla \varphi \wedge \nabla \psi |=1.$$

The two-variable density is $\rho (s,r)=1$ on ${(0,1)}^2$. Hence

$$\left(G_{\rho }h\right)\left(r\right)={\int }_0^{1}h\left(s\right)ds,$$

which is a rank-one operator. Therefore $S=C_J^I{G}_{\rho }$ is compact. This example shows explicitly how a pointwise singular kernel becomes compact after transversal coarea averaging.

Example 5.3 

(A diagonal model with nonconstant weight). Let $\Omega ={(0,1)}^2$ and

$$\varphi (x_1,x_2)=\psi (x_1,x_2)=x_1+x_2.$$

Then the essential range may be taken as $I=[0,2]$. Since $|\nabla \varphi |=\sqrt{2}$, the density $a_{\varphi }$ is the length of the line segment $x_1+x_2=s$ inside the square divided by $\sqrt{2}$. Hence

$$a_{\varphi }\left(s\right)=\left\{\begin{array}{cc}s,\hfill & 0<s<1,\hfill \\ 2-s,\hfill & 1<s<2.\hfill \end{array}\right.$$

The reduced equation is therefore

$$\lambda h\left(t\right)-p.v.{\int }_0^2{\displaystyle \frac{a_{\varphi }\left(s\right)h\left(s\right)}{t-s}}ds=B_{\varphi }f\left(t\right).$$

This example illustrates why endpoint behavior matters in the diagonal case: the coefficient vanishes at the endpoints, and a complete Fredholm statement requires the weighted finite-Hilbert-transform theory.

Example 5.4 

(A degenerate non-transversal case outside the compact theory). Let $\varphi (x_1,x_2)=x_1$ and $\psi (x_1,x_2)=x_1^2$. The variables ϕ and ψ are functionally dependent, since $\psi ={\varphi }^2$. The map $(\varphi ,\psi )$ has rank one rather than rank two, so the transversal compactness mechanism does not apply. This type of example belongs to the remaining degenerate theory and should be treated by different singular-integral methods.

6. Discussion

The results show that, in the regular cases treated here, solvability of (1) is governed by the geometry of the scalar pair $(\varphi ,\psi )$. When the ranges are separated, the equation is regular after coarea reduction and compact Fredholm theory applies. When the ranges overlap transversally, the kernel remains singular pointwise, but the two-variable coarea decomposition introduces an averaged operator $G_{\rho }$. If $\rho$ is Hölder continuous, this averaged operator is compact into $L^p$, and the finite Cauchy transform does not destroy compactness. In contrast, when $\psi =\varphi$, no transversal averaging occurs; the singularity persists and the reduced equation is a weighted finite Hilbert transform equation. The Fredholm transfer lemma shows that the reduced operator is not merely a formal device: for every nonzero spectral parameter, Fredholmness and compatibility conditions of the full product operator and of the reduced product operator are equivalent.

The theory is intentionally partial. It does not cover arbitrary degenerate pairs $(\varphi ,\psi )$, nor does it fully treat boundary singularities of the coarea densities. The following cases remain outside the compact framework developed here:

(i)

non-transversal intersections for which $rankD(\varphi ,\psi )<2$ on part of the interaction set;

(ii)

functionally dependent pairs $\psi =\eta \circ \varphi$ with nonlinear $\eta$, especially when ${\eta }^{\prime }$ vanishes;

(iii)

endpoint degeneracies where $a_{\varphi }$, $a_{\psi }$, or $\rho$ fail to be bounded or continuous;

(iv)

weighted $L^p$-settings adapted to vanishing or singular coarea densities;

(v)

a complete Fredholm index theory for the diagonal and more general dependent cases.

Thus the present paper should be read as a rigorous coarea-Fredholm framework for geometrically regular cases of Maz’ya’s multidimensional Cauchy-type equation, not as a complete solution of the general open problem. Its main value is to identify exactly when the multidimensional singularity can be transferred to a compact or classical one-dimensional operator, and to mark the degeneracies where that transfer no longer supplies compact Fredholm theory.

7. Conclusions

We have developed a partial coarea-based solvability theory for the multidimensional Cauchy-type equation

$$\lambda u\left(x\right)-p.v.{\int }_{\Omega }{\displaystyle \frac{u\left(y\right)}{\varphi \left(x\right)-\psi \left(y\right)}}dy=f\left(x\right).$$

The main conclusions are as follows.

(i)

Under explicit boundedness assumptions on the coarea densities $a_{\varphi }$ and $a_{\psi }$, the multidimensional operator factors as $K=A_{\varphi }{B}_{\psi }$.

(ii)

For $\lambda \ne 0$, the multidimensional equation is equivalent to the reduced one-dimensional equation $\lambda h-Sh=B_{\psi }f$, with reconstruction $u=(f+A_{\varphi }h)/\lambda$.

(iii)

If $\left(\varphi \right(\Omega ),\psi (\Omega \left)\right)>0$, then S is compact and the Fredholm alternative applies.

(iv)

If the singularity is transversal and the two-variable coarea density $\rho$ is Hölder continuous, then $S=C_J^I{G}_{\rho }$ is compact and the Fredholm alternative again applies.

(v)

If $\psi =\varphi$, then the reduced equation is a weighted finite Hilbert transform equation, and solvability is governed by Noether theory for one-dimensional singular integral operators.

These results provide a rigorous contribution to the solvability theory of Maz’ya-type multidimensional Cauchy integral equations under restrictive geometric hypotheses and identify the precise mechanisms responsible for compact Fredholm behavior. They also separate the compact regimes from the diagonal finite-Hilbert-transform regime and from genuinely degenerate configurations that require further analysis.