arXiv subjects

Tuomas Oikari

Publications and source records attributed to Tuomas Oikari.

15 recordsLinked to original sources

arXiv2026

Bounds for the maximal and Riesz potential operators with variable fractionality

We prove $L^{p(\cdot)}$-to-$L^{q(\cdot)}$ bounds for variable versions of the fractional maximal $M^{\alpha(\cdot)}$ and Riesz potential $I^{\alpha(\cdot)}$ operators. The changing fractionality in these operators is given by averaging the function $\alpha(\cdot)$ over balls. The bounds for $M^{\alpha(\cdot)}$ are in terms of a three-exponent Muckenhoupt condition relating $p(\cdot),q(\cdot),$ and $\alpha(\cdot)$, while the bounds for $I^{\alpha(\cdot)}$ are in terms of the boundedness of $M^{\alpha(\cdot)}$ and a packing condition on $\alpha(\cdot).$ These bounds hold under Hardy--Littlewood maximal function boundedness and Muckenhoupt conditions on the individual exponents $p(\cdot),q(\cdot),\alpha(\cdot).$ The proofs are based on an adaptation of sparse domination to variable fractionality and an embedding into variable sequential spaces.

math.CA↗

arXiv2025

Domains with globally exponentially integrable parabolic forward-in-time BMO

We characterize those open sets of the space time in which parabolic forward-in-time BMO functions are in a certain forward-in-time exponential integrability class. The characterization holds under qualitative connectivity assumptions on the domain and is formulated in terms of a quantitative growth bound on a forward-in-time version of the classical quasihyperbolic distance.

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arXiv2025

Factorizations for variable exponent Muckenhoupt weights

Given two variable exponent Muckenhoupt weights $w\in \mathcal{A}_{p(\cdot)}$ and $w_1\in \mathcal{A}_{p_1(\cdot)}$, we prove that for all small enough $\theta>0,$ there holds that $w_0\in \mathcal{A}_{p_0(\cdot)},$ where the weight is determined by $w = w_0^{1-\theta}w_1^{\theta}$ and exponent of the weight class by $1/p(\cdot) = (1-\theta)/p_0(\cdot) + \theta/p_1(\cdot).$ The proof is based on a recent reverse H\"older's inequality for variable exponent Muckenhoupt weights by Cruz-Uribe and Penrod. We upgrade these factorizations to the restricted range context by using a recent transformation formula due to Nieraeth. Then, following an extrapolation of compactness scheme by Hyt\"onen and Lappas, we provide an alternative proof of the recent extrapolation of compactness results of Lorist and Nieraeth in the context of weighted variable exponent Lebesgue spaces.

math.FA↗

arXiv2024

On the compactness of the bi-commutator

We prove compactness results and characterizations for the bi-commutator $[T_1,[b, T_2]]$ of a symbol $b$ and two non-degenerate Calderón-Zygmund singular integral operators $T_1, T_2$. Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm $L^{p_1}L^{p_2}\to L^{q_1}L^{q_2}$ off-diagonal case with $p_i < q_i$, and then extend these to other exponents, including the diagonal $p_i = q_i$, with a new extrapolation argument. In particular, the natural product $\operatorname{VMO}$ condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality $p_i \le q_i$ is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama.

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arXiv2024

Curved commutators in the plane

We complete the $L^p$ boundedness theory of commutators of Hilbert transforms along monomial curves by providing the previously missing lower bounds. This optimal result now covers all monomial curves while the previous result assumed the curve to intersect adjacent quadrants of the plane. We also develop, under a qualitative $\mathrm{BMO}$ assumption of the symbol, the corresponding quantitative lower bound in the context of curves with non-vanishing torsion.

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arXiv2024

Note on the compactness of commutators

The optimal sufficient conditions for the $L^p$-to-$L^q$ compactness of commutators of singular integral operators of both Calder\'on-Zygmund and of rough type are shown in the different exponent ranges $``q>p"$, $``q=p"$ and $``q<p"$ to quickly follow from each other. The approach is through classical compactness interpolation methods. We also present a new elementary off-diagonal to diagonal extrapolation principle for the compactness of commutators of linear operators, which is of independent interest.

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arXiv2024

Traces of vanishing H\"older spaces

For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the H\"older space $\dot{C}^{0,\omega}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/\omega(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of H\"older functions $\dot{C}^{0,\omega}(E)$ by Lipschitz and boundedly supported functions.

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arXiv2024

Approximation in H\"older Spaces

We introduce new vanishing subspaces of the homogeneous H\"{o}lder space $\dot{C}^{0,\omega}(X,Y)$ in the generality of a doubling modulus $\omega$ and normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing subspaces to completely characterize those H\"older functions that admit approximations, in the H\"{o}lder seminorm, by smooth, Lipschitz and boundedly supported functions. We present connections to bi-parameter harmonic analysis on the Euclidean space by providing applications to the compactness of the bi-commutator of two Calder\'on-Zygmund operators

math.FA↗

arXiv2023

On the $L^p$-to-$L^q$ boundedness and compactness of commutators along monomial curves

For exponents $p,q\in (1,\infty),$ we study the $L^p$-to-$L^q$ boundedness and compactness of the commutator $[b,H_γ] = bH_γ - H_γb,$ where $H_γ$ is the Hilbert transform along the monomial curve $γ$ and the function $b$ is complex valued. We obtain a sparse form domination of the commutator and show that $b\in \operatorname{BMO}^{γ,α}$ is a sufficient condition for boundedness in a half-open range of exponents $q\in[p,p+p(Ξ));$ in doing so we obtain a genuinely fractional case $q>p$ and also give a new proof of the case $q=p,$ due to Bongers, Guo, J. Li and Wick. Involving the $L^p$ improving phenomena of single scale averages, we show that $b\in \operatorname{VMO}^{γ,α}$ is a sufficient condition for compactness in the same range of exponents $q\in[p,p+p(Ξ))$ as in the case of boundedness. For $q<p,$ we show that $b\in \dot L^r$ is a sufficient condition for compactness; the argument is new and we use it to obtain a new short proof of the same implication for commutators of Calderón-Zygmund operators, recently due to Hytönen, K. Li, Tao and Yang. For necessity, and now restricting to the plane, we develop the approximate weak factorization argument to the extent that it now works for all monomial curves, thus removing the extra assumption present in our earlier work that the graph $γ(\mathbb{R})\subset\mathbb{R}^2$ intersects adjacent quadrants. As a consequence in the plane, we obtain that all of the above sufficiency conditions are necessary. We present several open problems.

math.CA↗

arXiv2023

Fractional Bloom boundedness and compactness of commutators

Let $T$ be a non-degenerate Calderón-Zygmund operator and let $b:\mathbb{R}^d\to\mathbb{C}$ be locally integrable. Let $1<p\leq q<\infty$ and let $μ^p\in A_p$ and $λ^q\in A_q,$ where $A_{p}$ denotes the usual class of Muckenhoupt weights. We show that \begin{align*} \|[b,T]\|_{L^p_μ\to L^q_λ}\sim \|b\|_{\operatorname{BMO}_ν^α},\qquad [b,T]\in \mathcal{K}(L^p_μ, L^q_λ)\quad\mbox{iff}\quad b\in \operatorname{VMO}_ν^α, \end{align*} where $L^p_μ=L^p(μ^p)$ and $α/d = 1/p-1/q,$ , the symbol $\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted $\operatorname{BMO}_ν^α$ and $\operatorname{VMO}_ν^α$ spaces are defined through the following fractional oscillation and Bloom weight \begin{align*} \mathcal{O}_ν^α(b;Q) = ν^{-α/d}(Q)\Big(\frac{1}{ν(Q)}\int_Q |b-\langle b\rangle_Q|\Big),\qquad ν = \big(\fracμλ\big)^β,\quad β= (1+α/d)^{-1}. \end{align*} The key novelty is dealing with the off-diagonal range $p<q$, whereas the case $p=q$ was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions $b$, while other arguments based on the median of $b$ on a set are inherently real-valued.

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arXiv2022

Off-diagonal estimates for commutators of bi-parameter singular integrals

We study the boundedness of commutators of bi-parameter singular integrals between mixed spaces $$ [b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2} $$ in the off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow $q_i\not= p_i.$ Boundedness is fully characterized for several arrangements of the integrability exponents with some open problems presented.

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arXiv2021

Lower bound of the parabolic Hilbert commutator

Answering a key point left open in a recent work of Bongers, Guo, Li and Wick, we provide the following lower bound $$ \|b\|_{\text{BMO}_γ(\mathbb{R}^2)}\lesssim \|[b,H_γ]\|_{L^p(\mathbb{R}^2)\to L^p(\mathbb{R}^2)}, $$ where $H_γ$ is the parabolic Hilbert transform.

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arXiv2021

Off-diagonal estimates for bilinear commutators

We find a minimal notion of non-degeneracy for bilinear singular integral operators $T$ and identify testing conditions on the multiplying function $b$ that characterize the $L^p\times L^q\to L^r,$ $1 \frac{1}{2},$ boundedness of the bilinear commutator $[b,T]_1(f,g) = bT(f,g) - T(bf,g).$ Our arguments cover almost all arrangements of the integrability exponents $p,q,r,$ with a single open problem presented in the end. Additionally, the arguments extend to the multilinear setting.

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arXiv2020

Off-diagonal estimates for bi-commutators

We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calderón-Zygmund operators, and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm integrability exponents $(p_1,p_2)\neq(q_1,q_2)$. The strategy is based on a bi-parameter version of the recent approximate weak factorization method.

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arXiv2019

Iterated commutators under a joint condition on the tuple of multiplying functions

We present a pair of joint conditions on the two functions $b_1,b_2$ strictly weaker than $b_1,b_2\in \operatorname{BMO}$ that almost characterize the $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ of these functions and a Calder\'on-Zygmund operator $T.$ Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.

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