Degenerated majorants of functions with zero ball means (Q2880014)

The functions having zero integrals over the balls of the fixed radius are studied. The main result includes sufficient conditions on the growth of the given function providing that it function belongs to the class mentioned above. More detail. Given \(r>0,\) denote by \(V_r({\mathbb R}^n)\) the set of all functions \(f\in L_{loc}^1({\mathbb R}^n)\) such that \(\int\limits_{| x| \leq r}f(x+y)dx=0\) for every \(y\in {\mathbb R}^n.\) Denote by \(\nu_n\) the smallest zero of the function \(J_{n/2}\) on \((0, +\infty),\) where \(J_{\mu}(z)\) denotes the Bessel function of the first type and of the order \(\mu.\) The main result of the paper is the following. Suppose that \(\theta\) is even continuous and positive function on \({\mathbb R}^1\) such that \(\int\limits_{0}^{\infty}\frac{ds}{\theta(s)}=\infty.\) Suppose also that a function \(\rho:{\mathbb R}^1\rightarrow {\mathbb R}^1\) satisfies the condition \(0<\rho(t)<\theta(\xi)\) at every \(t\in {\mathbb R}^1\) and \(\xi\in [t-\rho(t), t+\rho(t)].\) Then for every \(r>0\) there exists a non-zero function \(f\in V_r({\mathbb R}^n)\cap C^{\,\infty}({\mathbb R}^n)\) such that \(| f(x)| \leq c\inf\limits_{k\in {\mathbb Z}^+}\sqrt{k}\left(\frac{kr}{e}\right)^k \left(x_1^2+\cdots+x_{n-1}^2\right)^{\frac{-k}{2}}e^{\nu_n| x_n| /r}F_k(x_n),\) where \(c\) is some positive constant, and \(F_k(x_n)=\int\limits_{-\infty}^{\infty}(\rho(t))^{-k}\exp\left( \frac{-\pi}{2}\int\limits_0^t \frac{ds}{\theta(s)}+t+\rho(t)\frac{| x_n| }{r}\right)dt,\) \(x=(x_1,\ldots,x_n)\in {\mathbb R}^n.\) The results of the paper are applied to many problems of mathematical analysis and geometric function theory.