# Tagged Questions

Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic ...

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### an analogue of Littlewood-Paley-Rubio de Francia theory

For any function $f$ defined on the set of integer $\mathbb{Z}$, we define its Fourier transform as the following periodic function: $$\mathbb{F}f(\xi)=\sum_{n\in\mathbb{Z}}f(n)e^{-2\pi i n\xi}$$ ...
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### Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
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### Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
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### Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
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### Estimates for derivatives of a positive discrete harmonic function

There is the following estimation (Duffin, Discrete potential theory, Theorem 5): Let $f$ be a discrete harmonic function in a sphere of radius $R$ with the center $p$, all in $\mathbb Z^3$. Then, if ...
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### Proof of the chain rule for fractional derivatives in F.M.Christ and M.I.Weinsten's paper

In the paper: Dispersion of small amplitude Solutions of the generalized Korteweg-de Vries equation, JOURNAL OF FUNCTIONAL ANALYSIS 100,87-109 (1991). I find a mistake in the proof of proposition 3.1 ...
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### Stationary averages of harmonic measures

Let $A$ be a countable subset of the open unit disk $\mathbb D$ centered at 0. For a point $x\in\mathbb D$ denote by $\nu_x$ the associated harmonic measure on the boundary circle $\partial\mathbb D$. ...
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### extension of the projectivized gradient of a harmonic function

Let $(M,g)$ be a riemannian manifold, $\Delta$ the associated Laplacian, and $\{ f_i \}$ the real-valued eigenfunctions of $\Delta$. Then, $\nabla f_i \in \Gamma ^{\infty } (\mathrm{T} M)$ is defined ...
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### Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by $$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ By the classical Hardy-Littlewood-Sobolev theorem ...
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