# Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

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I am reading the following paper of Calderon and Zygmund http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm14122.pdf On page 3 (252) they provide an estimate (2.3) which is following $$|K(x-x_{\nu}) - K(-... • 800 1 vote 1 answer 30 views ### Harmonic conjugated functions and simply connected domain in wikipedia In wikipedia (link), it says: (i) Therefore, a harmonic function u admits a conjugated harmonic function if and only if the holomorphic function g(z)\colon=u_{x}(x,y)-iu_{y}(x,y) has a primitive ... • 171 0 votes 0 answers 41 views ### No Riemann-integrable function has the harmonic series has Fourier coefficients This is a question that was [already asked on this site][1] but got no satisfactory answer. I would like to rephrase and show my own attempt. So the point is, consider the sequence a_k = \begin{cases}... 1 vote 0 answers 26 views ### Explicite formula for Almansi decomposition In 1899 E. Almansi proved that any polyharmonic function of order p, i.e. in the kernel of the differential operator \Delta^p, defined on a shar shaped domain \Omega\subset\mathbb{R}^n, can be ... 2 votes 0 answers 36 views ### Reverse estimate of Hardy-Littlewood maximal function in Sobolev spaces Given f\in L^1_{loc}(\mathbb{R}^n), we consider the maximal function$$ Mf(x) := \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|dy. $$It is known that M : W^{1,p}(\mathbb{R}^n)\to W^{1,p}(\... • 219 0 votes 0 answers 20 views ### Formal Demonstration of an Integral Inequality Involving Nested Sets I am delving into an analysis involving a specific integral inequality that arises within the context of Fourier analysis, particularly focusing on expressions involving a series of integrations over ... 0 votes 0 answers 32 views ### Absolutely continuous measures perserve under a continuous function Let f: X \to \mathbb{R} be a continuous function over a compact metric space X. Assume that \mu and \nu are two Borel probability measures on X. Suppose that \mu << \nu. Is it true ... • 653 0 votes 0 answers 21 views ### Periodic Hilbert Transform for integrable functions is well-defined (Katznelson) I'm struggling to understand a proof in Katznelson (An Introduction to Harmonic Analysis, III - Lemma 1.3) Basically, given f \in L^1(T), take F(z) = e^{-f - i \tilde{f}}. Since this is ... 4 votes 1 answer 117 views ### Method to produce polyharmonic functions from harmonic ones Let F:\mathbb{R}^2\to\mathbb{R} be a harmonic function (it can be thought as one component of a holomorphic function), take m odd and let f:\mathbb{R}^{m+1}\to\mathbb{R},$$f(a,x_1,\dots,x_m)=\...
Assume $f\in \text{BMO}(\mathbb{R}^d)$. Denote $f_Q=\displaystyle\dfrac{1}{|Q|}\int_Q f(x)\,dx$. I want to show for $\alpha>2$ and any cube $Q$ with positive volume, we have \$|f_{\alpha Q}-f_Q|\leq ...