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Juha Kinnunen

Publications and source records attributed to Juha Kinnunen.

At least 19 recordsLinked to original sources
arXiv2026

Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations

We study parabolic Poincaré inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse Hölder inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincaré inequalities and self-improvement results on time intervals at fixed spatial points. These results are based on a pointwise maximal function estimate. As an application, we obtain regularity results in the time direction for solutions to nonlinear systems.

math.AP↗
arXiv2026

Doubling measures, Poincaré inequalities and parabolic Harnack inequalities for a doubly nonlinear equation

We characterize metric measure spaces satisfying parabolic Harnack inequalities for a doubly nonlinear equation in terms of volume doubling and Poincaré inequalities. Our approach uses purely analytical methods, based on obtaining estimates for solutions to a related Cauchy problem. This extends previous linear results to the nonlinear setting without relying on heat kernel estimates and representation formulae.

math.AP↗
arXiv2026

New inequalities related to sums of $L^p$ functions in connection with Carbery's problems

Carbery (2006) proposed novel estimates for the $L^p$ norm of a sum of two nonnegative measurable functions. Subsequently, Carlen, Frank, Ivanisvili and Lieb (2018) provided stronger bounds, which Ivanisvili and Mooney (2020) further refined to achieve estimates that are, in a certain sense, optimal. Continuing this line of research, the present work establishes new upper and lower bounds for the range \(p\in(1,\infty)\). Carbery also asked under what conditions on a sequence \((f_j)\) of nonnegative measurable functions the inequality \(\sum \|f_j\|_p^p < \infty\) implies that \(\sum f_j \in L^p\). Ivanisvili and Mooney (2020) resolved this question for \(p\in[1,2]\), and the present work proposes an answer for \(p\in[2,\infty)\).

math.FA↗
arXiv2025

Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term

We investigate a class of degenerate fully nonlinear nonlocal elliptic equations with Hamiltonian terms. By precisely characterizing the interaction between the degeneracy law of equations and the growth behavior of the Hamiltonian terms, we establish the Lipschitz regularity of viscosity solutions by the Ishii-Lions method, and further show the gradient Hölder continuity for solutions via utilizing perturbation techniques. Additionally, under minimal assumptions on the degeneracy pattern, the $C^1$-differentiability property of solutions is explored as well.

math.AP↗
arXiv2025

Self-improving properties of weighted norm inequalities on metric measure spaces

This work discusses self-improving properties of the Muckenhoupt condition and weighted norm inequalities for the Hardy-Littlewood maximal function on metric measure spaces with a doubling measure. Our main result provides direct proofs of these properties by applying a Whitney covering argument and a technique inspired by the Calderón-Zygmund decomposition. In particular, this approach does not rely on reverse Hölder inequalities.

math.CA↗
arXiv2024

Lipschitz truncation method for parabolic double-phase systems and applications

We discuss a Lipschitz truncation technique for parabolic double-phase problems of $p$-Laplace type in order to prove energy estimates and uniqueness results for the Dirichlet problem. Moreover, we show existence for a non-homogeneous double-phase problem. The Lipschitz truncation method is based on a Whitney-type covering result and a related partition of unity in the intrinsic geometry for the double-phase problem.

math.AP↗
arXiv2024

Characterizations of parabolic reverse Hölder classes

This paper discusses parabolic reverse Hölder inequalities and their connections to parabolic Muckenhoupt weights. The main result gives several characterizations for this class of weights. There are challenging features related to the parabolic geometry and the time lag, for example, in covering and chaining arguments. We also prove a Gehring type self-improving property for parabolic reverse Hölder inequalities.

math.CA↗
arXiv2024

Characterizations of parabolic Muckenhoupt classes

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of $p$-Laplace type. The main results include characterizations for the limiting parabolic $A_\infty$ and $A_1$ classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón-Zygmund decompositions, covering and chaining arguments appear in the proofs.

math.CA↗
arXiv2023

Gradient higher integrability for double phase problems on metric measure spaces

We study local and global higher integrability properties for quasiminimizers of a class of double-phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure space with a doubling measure and a Poincaré inequality. The main novelty is an intrinsic approach to double-phase Sobolev-Poincaré inequalities.

math.AP↗
arXiv2023

On the regularity theory for mixed anisotropic and nonlocal $p$-Laplace equations and its applications to singular problems

We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.

math.AP↗
arXiv2022

Parabolic Muckenhoupt Weights on Spaces of Homogeneous Type

This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse Hölder inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean oscillation and parabolic Muckenhoupt weights is studied by applying a parabolic John--Nirenberg lemma. A Coifman--Rochberg-type characterization of the space of parabolic bounded mean oscillation in terms of parabolic maximal functions is also given. The main challenges in the parabolic theory are related to the time lag in the estimates. The results are motivated by the corresponding Euclidean theory and the regularity theory for parabolic variational problems on metric measure spaces.

math.AP↗
arXiv2022

John-Nirenberg inequalities for parabolic BMO

We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John-Nirenberg inequality. Moreover, we consider John-Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition.

math.CA↗
arXiv2022

Variational solutions to the total variation flow on metric measure spaces

We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincaré inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.

math.AP↗
arXiv2021

Characterizations of weak reverse Hölder inequalities on metric measure spaces

We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak $A_\infty$ weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt $A_\infty$ weights that fail to hold for weak $A_\infty$ weights.

math.CA↗
arXiv2021

On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation.

math.AP↗
arXiv2021

On the definition of solution to the total variation flow

We show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals.

math.AP↗
arXiv2021

Dyadic John-Nirenberg space

We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John-Nirenberg space and provide a method to construct nontrivial functions in the dyadic John-Nirenberg space. Moreover, we prove that the John-Nirenberg space is complete. Several open problems are also discussed.

math.FA↗