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Bounded minimizers of double phase problems at nearly linear growth have locally H\"older continuous gradient within the sharp maximal nonuniformity range $q<1+\alpha$., Comment: 46 pages, comments are welcome

We deal with the following wide class of kinetic equations, $$ \big[\partial_t \,+\, v\cdot\nabla_x \big] f \, = \, L_v f. $$ Above, the diffusion term $L_v$ is an integro-differential operator, whose nonnegative kernel is of fractional order $s\in(0,1)$ having merely measurable coefficients. Firstly, we obtain a general $L^\infty$-interpolation inequality with a natural nonlocal tail term in velocity, in turn giving local boundedness even for weak subsolutions $f$ without any sign assumption. This is a veritable novelty, being boundedness usually assumed apriori in such a setting. Then, provided that their nonlocal tail in velocity is $(2\!+\!\varepsilon)$-summable along the transport variables, we prove a general Strong Harnack inequality, which in the simpler case of globally nonnegative weak solutions $f$ reads as follows $$ \sup_{Q^-} f \ \leq \ c\,\inf_{Q^+} f \,+ \, c\,\| {\textrm {Tail}} (f)\|_{L^{2+\varepsilon}_{t,x}}\,, $$ where $Q^{\pm}$ are suitable slanted cylinders. This is the first strong Harnack inequality for kinetic integral equations under the aforementioned tail summability assumption, which is in fact naturally implied in literature, e.g., from the usual mass density boundedness (as for the Boltzmann equation without cut-off), and in clear accordance with the very recent surprising counterexample by Kassmann and Weidner; see arXiv:2405.05223. A new standalone result, a Besicovitch-type covering argument for very general kinetic geometries, independent on the involved equation is also needed, stated and proved., Comment: We provide a new formulation of the strong Harnack inequality for kinetic integral equations, in accordance with Kassmann-Weidner's counterexample arXiv:2405.05223, and with the assumptions in the previous related literature; see Remark 1.3. Also, we removed global positiveness both in the weak and the strong Harnack inequalities, so that a natural nonlocal term does appear in the estimates

Minima of the log-multiphase variational integral $$ w \mapsto \int_{\Omega} \left[|Dw|\log(1+|Dw|) + a(x)|Dw|^q + b(x)|Dw|^s\right] \, {\rm d}x\,, $$ have locally H\"older continuous gradient under sharp quantitative bounds linking the growth powers $(q,s)$ to the H\"older exponents of the modulating coefficients $a(\cdot)$ and $b(\cdot)$ respectively.

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $p$-growth of the type $$ w \mapsto \int \left[F(Dw)-f\cdot w\right]dx $$ feature almost everywhere $\mbox{BMO}$-regular gradient provided that $f$ belongs to the borderline Marcinkiewicz space $L(n,\infty)$.

We investigate some of the effects of the lack of compactness in the critical Folland-Stein-Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi's $\Gamma$-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe [37]., Comment: Final version. To appear in Adv. Calc. Var

We investigate some effects of the lack of compactness in the critical Sobolev embedding in the Heisenberg group., Comment: arXiv admin note: substantial text overlap with arXiv:2307.14933

We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order $(s,p)$, with summability exponent $p \in (1,\infty)$ and differentiability exponent $s\in (0,1)$, whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

We investigate some effects of the lack of compactness in the critical Sobolev embedding by proving that a famous conjecture of Brezis and Peletier \cite{BP89} does still hold in the Heisenberg framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at exactly one point which is a critical point of the Robin function (i. e., the diagonal of the regular part of the Green function associated to the involved domain), in clear accordance with the underlying sub-Riemannian geometry. Consequently, a new suitable definition of domains geometrical regular near their characteristic set is introduced. In order to achieve the aforementioned result, we need to combine proper estimates and tools to attack the related CR Yamabe equation with novel feasible ingredients in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as for instance a fine asymptotic control of the optimal functions via the Jerison and Lee extremals realizing the equality in the critical Sobolev inequality \cite{JL88}., Comment: The most important addition in such a revision is the proof of the connection with the concentration point and the Robin function, in clear accordance with the analog in the Euclidean framework; see Thm. 1.2 at Page 5

After carrying out an overview on the non Euclidean geometrical setting suitable for the study of Kolmogorov operators with rough coefficients, we list some properties of the functional space $\mathcal{W}$, mirroring the classical $H^1$ theory for uniformly elliptic operators. Then we provide the reader with the proof of a new Sobolev embedding for functions in $\mathcal{W}$. Additionally, after reviewing recent results regarding weak regularity theory, we discuss some of their recent applications to real life problems arising both in Physics and in Economics. Finally, we conclude our analysis stating some recent results regarding the study of nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck operators.

We investigate local properties of weak solutions to nonlocal and nonlinear kinetic equations whose prototype is given by $$ \partial_t u +v\cdot\nabla_x u +(-\Delta_v)^s_p u = f(u). $$ We consider equations whose diffusion part is a (possibly degenerate) integro-differential operator of differentiability order $s \in (0,1)$ and summability exponent $p\in (1,\infty)$. Amongst other results, we provide an explicit local boundedness estimate by combining together a suitable Sobolev embedding theorem and a fractional Caccioppoli-type inequality with tail. For this, we introduce in the kinetic framework a new definition of nonlocal tail of a function and of its related tail spaces, also by establishing some useful estimates for the tail of weak solutions. Armed with the aforementioned results we give a precise control of the long-range interactions arising from the nonlocal behaviour of the involved diffusion operator.

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $p=2$, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent $s$ goes to $1$., Comment: We have removed the bound on the integrability exponent p. To appear in Calc. Var. Partial Differential Equations. arXiv admin note: text overlap with arXiv:2207.03741

We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and H\"older continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron-Wiener-Brelot generalized solution by proving its existence for very general boundary data., Comment: We have removed the bound on the integrability exponent p. To appear in Nonlinear Anal

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl group $\mathbb{H}^n$. Amongst other results, we prove that the weak solutions to such a class of problems are bounded and H\"older continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates., Comment: We have removed the bound on the integrability exponent p. To appear in J. Geom. Anal

We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.

In the framework of the Potential Theory we prove existence and uniqueness for the Perron-Wiener-Brelot solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for boundary regularity. Our results apply to a wide family of strongly degenerate operators that includes the following example~$\l = t^2\Delta_x + \langle x, \nabla_y \rangle -\partial_t$, for~$(x,y,t) \in \R^N \times \R^{N} \times \R$., Comment: 35 pages