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Bae Jun Park

Publications and source records attributed to Bae Jun Park.

At least 19 recordsLinked to original sources
arXiv2026

Weighted estimates for Multilinear Singular Integrals with Rough Kernels

We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_Ω$ associated with an integrable function $Ω$ on the unit sphere $\mathbb{S}^{mn-1}$ satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that $\mathcal{L}_Ω$ is bounded from $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)$ to $L^p(v_{\vec{\boldsymbol{w}}})$ under the assumption that $Ω\in L^q(\mathbb{S}^{mn-1})$ and that the $m$-tuple of weights $\vec{\boldsymbol{w}}= (w_1,\ldots,w_m)$ lies in the multiple weight class $\mathrm{A}_{\vec{\boldsymbol{p}}/q'}$. Here, $q'$ denotes the Hölder conjugate of $q$, and we assume $q'\le p_1,\dots,p_m<\infty$ with $1/p = 1/p_1 + \cdots + 1/p_m$.

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arXiv2026

Weighted Extensions of Stein's Theorem for Linear and Multilinear Operators

We study weighted estimates for linear and multilinear integral operators whose kernels satisfy only size conditions. Extending a theorem of E. Stein and its refinement by Soria and Weiss, we prove weighted estimates on Herz and Cesàro type spaces, together with multilinear strong-type and weak-type analogues. As applications, we derive consequences for a range of rough singular integral operators and related variants, including linear, oscillatory, and multilinear settings.

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arXiv2026

Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria

We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^α$. This improves the previously best known condition for boundedness of such bilinear operators obtained in the paper of the first and third authors, and provides estimates close to the conjectured endpoint of integrability suggested by the linear theory. The proof is based on a new sharp boundedness criterion for bilinear Fourier multiplier operators associated with sums of dyadic dilations of a fixed symbol $m_0$, compactly supported away from the origin. This criterion admits the best possible behavior with respect to a modulation of $m_0$ and is intimately connected with sharp shifted square function estimates.

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arXiv2026

Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators

We study optimal multiple weight assumptions in the weighted theory of multilinear Fourier multipliers and multilinear pseudo-differential operators. For multilinear Fourier multipliers, we revisit the weighted Hörmander-type theorem of Li and Sun, as a multilinear version of Kurtz and Wheeden, and show that their multiple weight condition is sharp. This provides the sharp necessary condition in the multilinear setting and simultaneously improves the classical linear necessity established by Kurtz and Wheeden. In the pseudo-differential setting, we consider recent weighted estimates of the authors for symbols in the multilinear Hörmander class and prove that their multiple weight hypothesis is also best possible. As a corollary, we can obtain the optimality of sharp maximal function estimates for multilinear pseudo-differential operators in the papers of the authors which originated from the results of Chanillo and Torchinsky.

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arXiv2025

Sharp bilinear estimates for maximal singular integrals with kernels in weighted $L^q$ spaces

In this paper, we study the boundedness properties of the (dyadic) maximal bilinear operator associated with rough homogeneous kernels on $\mathbb{R}$. We establish sharp $L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \to L^{p}(\mathbb{R})$ estimates in the full quasi-Banach range of exponents $1 < p_1, p_2 < \infty$ and $1/2 < p < \infty$. Our approach extends and unifies several recent contributions, including those of Honzík, the first author, and Slavíkova, as well as the second author in the bilinear and in the one-dimensional settings, by allowing the angular component $Ω$ of the kernel to belong to weighted $L^q$-spaces on $\mathbb{S}^1$.

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arXiv2025

Multilinear estimates for maximal rough singular integrals

In this work, we establish $L^{p_1}\times \cdots\times L^{p_m}\to L^p$ bounds for maximal multi-(sub)linear singular integrals associated with homogeneous kernels $\frac{Ω(\vec{\boldsymbol{y}}')}{|\vec{\boldsymbol{y}}|^{mn}}$ where $Ω$ is an $L^q$ function on the unit sphere $\mathbb{S}^{mn-1}$ with vanishing moment condition and $q>1$. As an application, we obtain almost everywhere convergence results for the associated doubly truncated multilinear singular integrals.

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arXiv2024

Sharp maximal function estimates for linear and multilinear pseudo-differential operators

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González, which provides more elaborate and natural weighted estimates in the multilinear setting.

math.AP↗
arXiv2024

Sharp Maximal function estimates for Multilinear pseudo-differential operators of type (0,0)

In this paper, we study sharp maximal function estimates for multilinear pseudo-differential operators. Our target is operators of type (0, 0) for which a differentiation does not make any decay of the associated symbol. Analogous results for operators of type (ρ, ρ), 0 < ρ< 1, appeared in an earlier work of the authors, but a different approach is given for ρ= 0

math.AP↗
arXiv2024

Vector-valued estimates for shifted operators

Shifted variants of (dyadic) Hardy-Littlewood maximal function and Stein's square function have played a significant role in the study of many important operators such as Calderon commutators, (bilinear) Hilbert transforms, multilinear multipliers, and multilinear rough singular integrals. Estimates for such shifted operators have a certain logarithmic growth in terms of the shift factor, but the optimality of the logarithmic growth has not yet been fully resolved. In this article, we provide sharp vector-valued shifted maximal inequality for generalized Peetre's maximal function, from which improved estimates for the above shifted operators follow with optimal logarithmic growths in a new way. We also obtain a vector-valued maximal inequality for the shifted (dyadic) Hardy-Littlewood maximal operator.

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arXiv2023

Trilinear Fourier multipliers on Hardy spaces

In this paper, we obtain the $H^{p_1}\times H^{p_2}\times H^{p_3}\to H^p$ boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calderón and Torchinsky (Adv. Math. 24 : 101-171, 1977). Our result improves the trilinear estimate in the very recent work of the authors, Lee, Heo, Hong, Park, and Yang (Math. Ann., to appear ) by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space $H^p$ for $0<p\le 1$.

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arXiv2023

A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators

In this paper, we investigate the Hörmander type theorems for the multi-linear and multi-parameter Fourier multipliers. When the multipliers are characterized by $L^u$-based Sobolev norms for $1<u\le 2$ , our results on the smoothness assumptions are sharp in the multi-parameter and bilinear case. In the multi-parameter and multi-linear case, our results are almost sharp. Moreover, even in the one-parameter and multi-linear case, our results improve earlier ones in the literature.

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arXiv2022

Improved estimates for bilinear rough singular integrals

We study bilinear rough singular integral operators $\mathcal{L}_Ω$ associated with a function $Ω$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slavíková (Math. Ann. 376: 431-455, 2020), they showed that $\mathcal{L}_Ω$ is bounded from $L^2\times L^2$ to $L^1$, provided that $Ω\in L^q(\mathbb{S}^{2n-1})$ for $4/3<q\le \infty$ with mean value zero. In this paper, we provide a generalization of their result. We actually prove $L^{p_1}\times L^{p_2}\to L^p$ estimates for $\mathcal{L}_Ω$ under the assumption $$Ω\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for }~\max{\Big(\;\frac{4}{3}\;,\; \frac{p}{2p-1} \;\Big)<q\le \infty}$$ where $1<p_1,p_2\le\infty$ and $1/2<p<\infty$ with $1/p=1/p_1+1/p_2$ . Our result improves that of Grafakos, He, and Honzík (Adv. Math. 326: 54-78, 2018), in which the more restrictive condition $Ω\in L^{\infty}(\mathbb{S}^{2n-1})$ is required for the $L^{p_1}\times L^{p_2}\to L^p$ boundedness.

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arXiv2022

Multilinear rough singular integral operators

We study $m$-linear homogeneous rough singular integral operators $\mathcal{L}_Ω$ associated with integrable functions $Ω$ on $\mathbb{S}^{mn-1}$ with mean value zero. We prove boundedness for $\mathcal{L}_Ω$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots, p_m<\infty$ and $1/p=1/p_1+\cdots +1/p_m$ in the largest possible open set of exponents when $Ω\in L^q(\mathbb S^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in $\mathbb R^m$.

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arXiv2022

On pointwise a.e. convergence of multilinear operators

In this work we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited decay. The a.e. convergence is deduced from the $L^2\times\cdots\times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.

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arXiv2022

On the boundedness of multilinear Fourier multipliers on Hardy spaces

In this paper, we study multilinear Fourier multiplier operators on Hardy spaces. In particular, we prove that the multilinear Fourier multiplier operator of Hörmander type is bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $H^p$ for $0<p_1,\dots,p_m\le 1$ with $1/p_1 + \cdots 1/p_m = 1/p$, under suitable cancellation conditions. As a result, we extend the trilinear estimates of the authors(arXiv:2107.00225) to general multilinear ones and improve the boundedness result of the authors, Lee, Heo, Hong, Park, and Yang(Math. Ann. 381 : 499-555, 2021) in limiting situations.

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arXiv2021

Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.

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arXiv2021

On the failure of multilinear multiplier theorem with endpoint smoothness conditions

We study a multilinear version of Hörmander multiplier theorem, namely \begin{equation*} \Vert T_σ(f_1,\dots,f_n)\Vert_{L^p}\lesssim \sup_{k\in\mathbb{Z}}{\Vert σ(2^k\cdot,\dots,2^k\cdot)\widehat{ϕ^{(n)}}\Vert_{L^{2}_{(s_1,\dots,s_n)}}}\Vert f_1\Vert_{H^{p_1}}\cdots\Vert f_n\Vert_{H^{p_n}}. \end{equation*} We show that the estimate does not hold in the limiting case $\min{(s_1,\dots,s_n)}=d/2$ or $\sum_{k\in J}{({s_k}/{d}-{1}/{p_k})}=-{1}/{2}$ for some $J \subset \{1,\dots,n\}$. This provides the necessary and sufficient condition on $(s_1,\dots,s_n)$ for the boundedness of $T_σ$.

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