This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence and uniqueness of maximal weak solutions in suitable weighted Z-spaces in the absence of smallness conditions. We showcase our theory with an application to rough reaction–diffusion equations. Subsequent articles will treat further classes of equations, including equations of Burgers-type and quasi-linear problems, using the same approach. Our toolkit includes a novel theory of hypercontractive singular integral operators (SIOs) on weighted Z-spaces and a self-improving property for super-linear reverse Hölder inequalities.

We introduce and study a new scale of function spaces that characterize the homogeneous Besov spaces Ḃ^β_p,q, hence completing earlier work by Ullrich. These new spaces include the ones introduced by Barton and Mayboroda, and systematically studied by Amenta under the name of weighted Z-spaces, for the purpose of boundary value problems with Ḃ^β_p,p data. They are the counterparts to the weighted tent spaces with Whitney averages, developed by Huang, and arise as their real interpolants. We describe their functional analytic properties: completeness, duality, embeddings, as well as their real and complex interpolants.

We establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. Moreover, we establish existence of bounded weak solutions when the initial data is merely bounded.

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse Hölder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.

We establish norm inequalities for fractional powers of degenerate Laplacians, with degeneracy being determined by weights in the Muckenhoupt class A_2(ℝ^n), accompanied by specific additional reverse Hölder assumptions. This extends the known results for classical Riesz potentials. The approach is based on size estimates for the degenerate heat kernels. The approach also applies to more general weighted degenerate operators.

We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this level of generality, and avoid any of the smoothness assumptions used in the literature, we introduce a notion of pathwise weak solution and develop a new harmonic analysis toolkit. The latter includes techniques to prove the boundedness of various maximal regularity operators on relevant spaces of square functions, the parabolic tent spaces $\mathrm{T}^{p}$. Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on $\mathrm{L}^{2}(\mathbb{R}_{+} \times \mathbb{R}^{n})=\mathrm{T}^2$ to $\mathrm{T}^p$, for all $1-\varepsilon

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational definition. Our classes of weak solutions are taken with minimal assumptions. We prove the existence and uniqueness of a fundamental solution which seems new in this generality: it is shown to always coincide with the associated evolution family for the initial value problem with zero source and it yields representation of all weak solutions. Our strategy is a variational approach avoiding density arguments, a priori regularity of weak solutions or regularization by smooth operators. One of our main tools are embedding results which yield time continuity of our weak solutions going beyond the celebrated Lions regularity theorem and that is addressing a variety of source terms. We illustrate our results with three concrete applications : second order uniformly elliptic part with Dirichlet boundary condition on domains, integro-differential elliptic part, and second order degenerate elliptic part.

This contribution starts with an exchange between us on the way we met Guido and he influenced our mathematical lives. Then it is mainly a survey paper that illustrates this influence by describing different topics and their subsequent evolution after his seminal papers and courses. Our main thread is the notion of a space of homogeneous type. In the second section we describe how it became central in pluricomplex analysis and consider particularly the existence of weak factorization for spaces of holomorphic functions. In the last section, one revisits the construction of a basis of wavelets in a space of homogeneous type and the way it allows a Littlewood-Paley analysis.

Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius. As compared to earlier results, Ricci curvature is only assumed to be bounded from below, and the weight is only assumed to be locally in A^2. The Kato square root estimate is proved under this weaker assumption. On compact Lipschitz manifolds we prove solvability estimates for solutions to degenerate elliptic systems with not necessarily self-adjoint coefficients, and with Dirichlet, Neumann and Atiyah-Patodi-Singer boundary conditions.

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of p for which tempered distributions in homogeneous Hardy–Sobolev spaces Ḣ^s,p with regularity index s ∈ (-1,1) are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces Ḃ^s_p,p.

We show the existence and uniqueness of fundamental solution operators to Kolmogorov--Fokker--Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is, when the diffusion operator is of differential type, we prove L2 decay using Davies's method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's L2 - LO estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay.

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov–Fokker–Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings à la Lions and transfer-of-regularity results à la Bouchut–H'́ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

We propose a simple method to obtain semigroup representation of solutions to the heat equation using a local L-2 condition with prescribed growth and a boundedness condition within tempered distributions. This applies to many functional settings and, as an example, we consider the Koch and Tataru space related to BMO-1 initial data.

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem ∂ _t u - div A ∇ u = f, u(0)=0 , where the source f also lies in (different) weighted tent spaces, provided the complex coefficient matrix A is bounded, measurable, time-independent, and uniformly elliptic. To achieve this, we extend the theory of singular integral operators on tent spaces via off-diagonal estimates introduced by [8] to obtain estimates on solutions u, and also ∇ u , ∂ _t u , and div A ∇ u in weighted tent spaces, showing at the same time maximal regularity. Uniqueness follows from a different strategy using interior representation for weak solutions and boundary behavior.

In this chapter we establish the existence part in our main results on the Dirichlet and Regularity problems with $${\operatorname {H}}^p$$ -data, Theorems 1.1 and 1.2 . When the data f additionally belongs to $${\operatorname {L}}^2$$ , the (eventually unique) solution is given by the Poisson semigroup. Hence, we proceed in two steps: First, we establish the required semigroup estimates for data $$f \in a^{-1}({\operatorname {H}}^p \cap {\operatorname {L}}^2)$$ and $$f \in \dot {\operatorname {H}}^{1,p} \cap \operatorname {W}^{1,2}$$ , respectively. Second, we obtain existence of a solution by a density argument for the full class of data.

This chapter contains all necessary background on function spaces that will be used later on.

In this chapter, we establish the existence part of Theorem 1.3 , our main result on the Dirichlet problems with boundary data in Hölder spaces and BMO.

This chapter complements Chaps. 17 , 18 and 19 . We shall prove the uniqueness parts in Theorems 1.1 , 1.2 , 1.3 , and 1.4 . In Auscher and Egert (Anal PDE 13(6): 1605–1632, 2020), we developed a strategy to prove uniqueness for elliptic systems without regularity assumptions and with coefficients not necessarily in block form. We streamline the strategy in the case of the block system ℒu = 0 to obtain uniqueness of solutions in much greater generality.