Alex Amenta, Gennady Uraltsev
We prove $L^p$-boundedness of variational Carleson operators for functions valued in intermediate UMD spaces. This provides quantitative information on the rate of convergence of partial Fourier integrals of vector-valued functions. Our proof relies on bounds on wave packet embeddings into outer Lebesgue spaces on the time-frequency-scale space $\mathbb{R}^3_+$, which are the focus of this paper.
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Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions
本文利用Bessel卷积与$\gamma$-radonifying范数刻画取值于UMD Banach空间的BMO奇函数空间，并证明UMD Banach空间是使Bessel-Poisson积分满足特定Carleson不等式的唯一空间。
We prove modulation invariant embedding bounds from Bochner spaces $L^p(\mathbb{W};X)$ on the Walsh group to outer-$L^p$ spaces on the Walsh extended phase plane. The Banach space $X$ is assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply $L^p$ bounds and sparse domination for the Banach-valued tritile operator, a discrete model of the Banach-valued bilinear Hilbert transform.
We prove the 'little Carleson theorem' on the growth of Fourier series for functions taking values in a UMD Banach space.
We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.
We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hytönen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q<r \leq \infty and (r/(q-1))' < p < \infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type t …
We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H]_θ$ between another UMD space $X$ and a Hilbert space $H$, for some $θ\in(0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.
We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.
In this paper we formulate embedding maps into time-frequency space related to the Carleson operator and its variational counterpart. We prove bounds for these embedding maps by iterating the outer measure theory of [DT15]. Introducing iterated outer $L^p$ spaces is a main novelty of this paper. [DT15] Yen Do and Christoph Thiele. "$L^p$ theory for outer measures and two themes of Lennart Carleson united". In: Bulletin of the American Mathematical Society 52.2 (2015), pp. 249-296.
In this paper, We study dimension-free $L^p$ estimates for UMD lattice-valued $q$-variations of Hardy-Littlewood averaging operators associated with the Euclidean balls.
Due to its nonlocal nature, the $r$-variation norm Carleson operator $C_r$ does not yield to the sparse domination techniques of Lerner, Di Plinio and Lerner, Lacey. We overcome this difficulty and prove that the dual form to $C_r$ can be dominated by a positive sparse form involving $L^p$ averages. Our result strengthens the $L^p$ estimates by Oberlin et. al. As a corollary, we obtain quantitative weighted norm inequalities improving on previous results by Do and Lacey. Our proof relies on the localized outer $L^p$-embeddings of Di Plinio-Ou and Uraltsev.
Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a UMD (Unconditional Martingale Differences) Banach space X. Every known UMD Banach space has finite quartile type, and it was recently shown that the Walsh analog of Carleson's Theorem holds under a closely related assumption of finite tile type. For a Walsh model of the bilinear Hilbert transform however, the quartile type should be sufficiently close to that of a Hilbert space for our results to hold. A full s…
We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.
In this paper, we establish UMD lattice-valued variational inequalities for differential operators, ergodic averages and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As a consequence, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence.
We prove variation and oscillation $L^p$-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational $L^p$-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish $L^p$-boundedness properties for weighted difference involving the semigroups under consideration.
Recently, in \cite{GXHTM}, the authors established $L^p$-boundedness of vector-valued $q$-variational inequalities for averaging operators which take values in the Banach space satisfying martingale cotype $q$ property. In this paper, we prove that martingale cotype $q$ property is also necessary for the vector-valued $q$-variational inequalities, which is a question left open. Moreover, we characterize UMD property and martingale cotype $q$ property in terms of vector valued $q$-variational inequalities for Hilbert transform.
In this paper, we establish jump and variational inequalities for the Calderón commutators, which are typical examples of non-convolution Calderón-Zygmund operators. For this purpose, we also show jump and variational inequalities for para-products and commutators from pseudo-differential calculus, which are of independent interest. New ingredients in the proofs involve identifying Carleson measures constructed from sequences of stopping times, in addition to many Littlewood-Paley type estimates with gradient.
A variation norm Carleson theorem in higher dimensions
本文将Fefferman的多边形傅里叶部分和极大函数界推广到高维下的强$r$-变分范数界与2-振荡界，并针对嵌套矩形的部分傅里叶和建立了相关变分与振荡界，完善了高维Carleson定理的研究。
The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper…
Sharp variational estimates of Stein-Wainger type operators
本文研究了Stein-Wainger型算子的$r$-变分在$ L^p(\R^n) $中的不等式，证明了其在端点处的锐利性，并解决了Guo-Roos-Yung提出的问题。
For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result of Hunt-Young and is a weighted extension of a variation norm Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lepingle.
A Variation Norm Carleson Theorem Along the Primes
研究沿素数的离散变分卡尔松算子及素数卡尔松算子在\(\ell^p\)上的有界性，通过结合高阶傅里叶均匀性等方法，给出处理调制不变奇异积分新机制，确定算子有界的\(p\)范围并得到非变分端点尖锐结果。
$q$-variational H{ö}rmander functional calculus and Schr{ö}dinger and wave maximal estimates
本文在扇形算子满足Hörmander泛函演算的条件下，证明m(tA)f对几乎处处变量具有有限q-变差（q>2），推广了半群情形；并进一步建立依赖时间的极大估计，应用于Schrödinger与波动方程传播子，得到Carleson型逐点收敛结果。
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