Higher order elliptic equations in weighted Banach spaces

We consider higher order linear, uniformly elliptic equations with non-smooth coefficients in Banach-Sobolev spaces generated by weighted general Banach Function Space (BFS). Supposing boundedness of the Hardy-Littlewood Maximal and Calderon-Zygmund singular operators in BFSs we obtain local solvability in the Sobolev-BFS and establish interior Schauder type a priori estimates for the. elliptic operator. These results will be used in order to obtain Fredholmness of the operator under consideration in weighted BFSs with suitable weight. In addition, we analyze some examples of weighted BFS that verify our assumptions and in which the corresponding Schauder type estimates and Fredholmness of the operator hold true.

💡 Research Summary

This paper investigates higher-order linear uniformly elliptic partial differential equations (PDEs) within the framework of weighted Banach Function Spaces (BFS). The authors consider an m-th order elliptic operator L with non-smooth coefficients, defined on a bounded domain Ω in R^n with smooth boundary. The core function space setting is a weighted BFS, denoted X_w(Ω), generated by a Banach function norm and a positive weight function w. The corresponding Sobolev-type space is W^m X_w(Ω).

The primary objective is to establish local solvability and interior Schauder-type a priori estimates for solutions to L u = f in these generalized spaces. The key analytical assumption (Property 1) is the boundedness of two fundamental operators in X_w(Ω): the Hardy-Littlewood maximal operator (M) and the Calderón-Zygmund singular integral operators (K). This assumption effectively generalizes the Calderón-Zygmund regularity theory to the abstract BFS setting. An additional embedding property (X_w ⊂ L^p for some p>1) and an extension property for the Sobolev-Banach spaces (Property 2) are also required.

The operator L is assumed to be uniformly elliptic, and its highest-order coefficients satisfy a local continuity condition, denoted (P_{x_0}), at a point x_0. This condition allows for the construction of a “tangential operator” L_{x_0} with constant (frozen) highest-order coefficients at x_0 and its fundamental solution (parametrix) J_{x_0}. Using this parametrix, any function φ can be represented as φ = T_{x_0} φ + S_{x_0} L φ, where S_{x_0} is the integral operator with kernel J_{x_0} and T_{x_0} = S_{x_0}(L_{x_0} - L).

The central technical result (Lemma 3.1, the Main Lemma) proves that for functions φ with compact support in a small ball B_r(x_0), the operator T_{x_0} acts as a contraction on the space W^m X_w(B_r) equipped with a suitable scaled norm. The proof meticulously handles lower-order derivatives using estimates related to the Riesz potential and the maximal function, and the highest-order derivatives using the boundedness of the singular integral operators assumed in Property 1. The contraction factor σ(r) tends to zero as the ball’s radius r decreases, relying on the local continuity (P_{x_0}) of the coefficients.

This contraction property is then leveraged to prove the main existence theorem (Theorem 4.3). For a given right-hand side f ∈ X_w, the equation L u = f is reformulated as a fixed-point problem u = T_{x_0} u + S_{x_0} f in W^m X_w(B_r). Since T_{x_0} is a contraction for sufficiently small r, the Banach fixed-point theorem guarantees the existence of a unique local solution u ∈ W^m X_w(B_r). Lemma 4.1 establishes the equivalence between this fixed-point condition and being a strong solution to the PDE.

The authors indicate that these local solvability results and the derived a priori estimates are fundamental for establishing the Fredholm property of the operator L acting between the weighted Sobolev-Banach space and the base BFS on the entire domain Ω. Finally, the paper provides examples of concrete weighted BFS that satisfy all the required properties, such as classical weighted Lebesgue spaces L^p_w with Muckenhoupt A_p weights, and variable exponent Lebesgue spaces L^{p(·)}_w, thereby demonstrating the applicability of their abstract theory to a wide range of non-standard function spaces important in modern PDE analysis.