This will be an expository talk which aims to introduce some problems in harmonic analysis and geometric measure theory concerning the geometry of a measure for which an associated integral operator is well behaved. As an example, we shall prove a result of Mattila and Preiss concerning the relationship between the rectifiability of a measure and the existence of the Riesz transform in the sense of principle value.
We present new results concerning characterizations of the spaces $C^{1,\alpha}$ and “$LI_{\alpha+1}$” for $0<\alpha<1$. The space $LI_{\alpha +1}$ is the space of Lipschitz functions with $\alpha$-order fractional derivative having bounded mean oscillation. These characterizations involve geometric square functions which measure how well the graph of a function is approximated by a hyperplane at every point and scale. We will also discuss applications of these results to higher-order rectifiability.
The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity").
The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator
$$
H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R
$$
is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1
I’ll present a quantitative version of a stability estimate
for the Sobolev Inequality improving previous results of Bianchi
and Egnell. The estimate has the correct dimensional dependence
which leads to a stability estimate for the Logarithmic Sobolev inequality.
This is joint work with Dolbeault, Esteban, Figalli and Frank.
The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n.
It is known that if $\{x_n\}$ is a frame for a separable Hilbert space, then there exist some sequences $\{y_n\}$ such that $x= \sum x_n$, and this sum converges in the norm of H. This equation is called the reconstruction formula of x. In this talk, we will talk about the existence of frames that admit absolutely convergent and unconditionally convergent reconstruction formula. Some characterizations of such frames will also be presented.
Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$.
