Knowlpedia

Proposition

Radial operator bounds in an analytic coefficient norm

Raising radial degree yields bounds for integration and for a parameter derivative followed by integration.

Statement

Let IYF=∫0YF(s,η) dsI_YF=\int_0^YF(s,\eta)\,ds in the . Its weights satisfy

wαβwα+1,β≤4R,wα,β+1(α+1)wα+1,β≤4Rρ.\frac{w_{\alpha\beta}}{w_{\alpha+1,\beta}}\le4R, \qquad \frac{w_{\alpha,\beta+1}}{(\alpha+1)w_{\alpha+1,\beta}}\le\frac{4R}{\rho}.

These follow by canceling the factorials and binomial coefficients in the definition.

Consequences

Multiplication by YY and IYI_Y each have norm at most 4R4R; the has norm at most one; and ∥∂ηIY∥≤4R/ρ\|\partial_\eta I_Y\|\le4R/\rho. For ν≥1\nu\ge1, if Fα=0F_\alpha=0 for α<b\alpha<b, then

∥JνF∥R,ρ≤4R(b+1)(b+ν)∥F∥R,ρ.\|J_\nu F\|_{R,\rho} \le\frac{4R}{(b+1)(b+\nu)}\|F\|_{R,\rho}.

This uses the coefficient divisor (α+1)(α+ν)(\alpha+1)(\alpha+\nu) of the . A bare parameter derivative need not be bounded in the same norm; integration supplies the missing radial degree.