Some estimates for the Jacobi transform in the space \(\operatorname{L}^2(\mathbb R^+,\Delta_{(\alpha,\beta)}(t)dt)\) (Q2906316)

The Jacobi function \(\phi^{(\alpha,\beta)}_\lambda(t)\) of order \((\alpha,\beta)\) \((\alpha\neq -1,-2,\dots)\) is the unique \(C^\infty\)-function on \(\mathbb R\) which equals 1 at 0 and satisfies the differential equation NEWLINE\[NEWLINE(D_{\alpha,\beta}+ \lambda^2+ \rho^2)\phi^{(\alpha, \beta)}_\lambda(t)= 0,NEWLINE\]NEWLINE where \(\lambda\in C\); \(\rho=\alpha+ \beta+ 1\) and NEWLINE\[NEWLINED_{\alpha,\beta}= {d^2\over dt^2}+ ((2\alpha+ 1)\text{coth\,}t+ (2\beta+ 1)\tanh t){d\over dt}.NEWLINE\]NEWLINE Consider the space \(L^2_{\alpha,\beta)}(\mathbb R^+)= L^2(\mathbb R^+, \Delta(\alpha, \beta)(t)\,dt)\) with the norm NEWLINE\[NEWLINE\| f\|=\| f\|_{2,(\alpha, \beta)}= \Biggl(\int^\infty_0 |f(x)|^2 \Delta_{(\alpha, \beta)}(x)\,dx\Biggr)^{{1\over 2}},NEWLINE\]NEWLINE where \(\Delta_{(\alpha, \beta)}(t)= (2\sinh t)^{2\alpha+ 1}\cdot (2\cosh t)^{2\beta+ 1}\).NEWLINENEWLINE The Jacobi transform of a function \(f\) from \(L^2_{(\alpha, \beta)}(\mathbb R^+)\) is defined by NEWLINE\[NEWLINEg(\lambda)= \int^\infty_0 f(t) \Phi^{(\alpha,\beta)}_\lambda(t) \Delta_{(\alpha,\beta)}(t)\,dt.NEWLINE\]NEWLINE In the paper, the next integral is estimated NEWLINE\[NEWLINE\int_{\lambda\geq N} |g(\lambda)|^2\,d\mu(\lambda),NEWLINE\]NEWLINE where \(d\mu(\lambda)= |C(\lambda)|^{-2}\,d\lambda\) with special \(C(\lambda)\).NEWLINENEWLINE Two useful estimates are proved in a certain class of functions characterized by a generalized continuity modulus, using a generalized translation operator.