arXiv subjects

Jin Bong Lee

Publications and source records attributed to Jin Bong Lee.

13 recordsLinked to original sources

arXiv2026

The strong $\ell^r$ spherical maximal function

Let $d\ge3$ and $2\le r<\infty$. We consider the strong maximal operator associated with Euclidean surface averages over the $\ell^r$ unit sphere and independent coordinate dilations. We prove that this operator is bounded on $L^p(\mathbb R^d)$ if and only if \[ p>\max\left\{\frac{d+1}{d-1},\frac{r}{d-1}\right\}. \] We also obtain $L^p\to L^q$ bounds for the corresponding local maximal operator. For each fixed $p$ in certain ranges, the resulting range of $q$ is optimal up to endpoints. The proof uses a dyadic surface decomposition and frequency-localized $L^2$ estimates. We establish these estimates by combining exact multiplier factorization via iterated stationary phase with scalar Fourier inversion and $TT^*$ arguments. In dimension three, these estimates are combined with quantitative multiparameter local smoothing.

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arXiv2026

Sharp $L^2$ Estimates for $(2+1)$-dimensional oscillatory integral operators with homogeneous binomial phases

We study oscillatory integral operators in $(2+1)$-dimensions with a homogeneous binomial phase \[ Φ(x,y,t)=x^{k-k_P}t^{k_P}+y^{k-k_Q}t^{k_Q}, \qquad 1\le k_P<k_Q<k. \] For compactly supported smooth amplitudes, we establish sharp $L^2(\R)\to L^2(\R^2)$ estimates with logarithmic losses occurring only in certain critical cases. The proof is based on scale-dependent Phong--Stein estimates.

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arXiv2026

Maximal averages and non-transversality

We investigate the $L^p$ mapping properties of maximal functions associated with analytic hypersurfaces in $\mathbb R^d$, with a particular emphasis on the role of transversality. Around points that are not transversal, we show that the associated maximal function is bounded on $L^p(\mathbb R^d)$ for all $p>2$, regardless of the decay of the Fourier transform of surface measures. In contrast, away from non-transversal points, we prove that $L^p$ bounds for the maximal operator imply that the Fourier transform of the surface measure decays at rate $1/q$ for $q>p$. Combining these two regimes, we demonstrate that the conjecture of Stein and Iosevich-Sawyer on maximal functions could be re-formulated, in the analytic setting, by restricting attention to transversal points. Moreover, our result completely settles the refined form of the conjecture for certain cases.

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arXiv2025

Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse Hölder inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.

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arXiv2025

Maximal operators given by Fourier multipliers with dilation of fractional dimensions

In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $L^p$ bounds of the maximal operators for each $p$. Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.

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arXiv2025

A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order.

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arXiv2025

Strichartz and local smoothing estimates for the fractional Schrödinger equations over fractal time

We obtain Strichartz-type estimates for the fractional Schrödinger operator $f \mapsto e^{it(-Δ)^{γ/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the notions in the spirit of the Assouad dimension, such as, bounded Assouad characteristic and Assouad specturm. We also prove the estimate $$ \| e^{it(-Δ)^{γ/2}} f \|_{L_t^q(\mathrm{d}μ; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$ where $μ$ is a measure satisfying an $α$-dimensional growth condition. In addition, we establish related inhomogeneous estimates and $L^2$ local smoothing estimates. A surprising feature of our work is that, despite dealing with rough fractal sets, we extend the known estimates for the fractional Schrödinger operators in a natural way, precisely consistent with the associated fractal dimensions.

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arXiv2024

$L^p$ improving properties and maximal estimates for certain multilinear averaging operators

In this article we focus on $L^{p}$ estimates for two types of multilinear lacunary maximal averages over hypersurfaces with curvature conditions. Moreover, we give a different proof for the bilinear lacunary spherical maximal functions. To obtain our results, we make use of the $L^1$-improving estimates of multilinear averaging operators. We also obtain $L^p$-improving estimates for certain multilinear averages by means of the nonlinear Brascamp-Lieb inequality.

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arXiv2024

Improved curvature conditions on $L^2\times\cdots\times L^2 \to L^{2/m}$ bounds for multilinear maximal averages

In this article, we focus on $L^{2}(\mathbb{R}^d)\times\cdots\times L^{2}(\mathbb{R}^d)\rightarrow L^{2/m}(\mathbb{R}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear averages with $m\geq3$, and represent a reproof of the recent result of T. Borges, B. Foster, and Y. Ou on the curvature conditions of the hypersurfaces required in establishing $L^{2}(\mathbb{R}^d)\times L^{2}(\mathbb{R}^d)\rightarrow L^{1}(\mathbb{R}^d)$ estimates of bilinear maximal functions.

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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

On the trace theorem to Volterra-type equations with local or non-local derivatives

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $L_p$ spaces with $A_p$ weight. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.

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arXiv2023

Maximal operators associated with Fourier multipliers and applications

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which are not necessarily radial. To do so, we make use of modified square function estimates and bilinear interpolation. In result, we obtain convergence results for fractional half-wave equations and surface averages as well as the $L^p$ boundedness for the maximal 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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