Fixed-regularization periodic PIC kernels are globally Lipschitz

By [F2], each grid weight is smooth and has bounded derivatives of every finite order. By [F1], ${\mathcal{E}}_{\epsilon ,h}$ is a finite sum of products of such weights and fixed matrix entries, so it is $C^{\mathrm{\infty }}$ and all its derivatives are bounded. In particular, both derivative suprema $M_x^{\epsilon ,h},M_y^{\epsilon ,h}$ are finite. More explicitly, if $B_{jk}$ are the vector entries of $-D^0{A}_h^{†}$ and $C_w={\max }_j\mathrm{\parallel }\mathrm{\nabla }{w}_j{\mathrm{\parallel }}_{\mathrm{\infty }}$, then $M_x^{\epsilon ,h}\le \mathrm{\mid }cQ\mathrm{\mid }{h}^{-d}{C}_w{\sum }_{j,k}\mathrm{\mid }{B}_{jk}\mathrm{\mid }\mathrm{\parallel }{w}_k{\mathrm{\parallel }}_{\mathrm{\infty }},M_y^{\epsilon ,h}\le \mathrm{\mid }cQ\mathrm{\mid }{h}^{-d}{C}_w{\sum }_{j,k}\mathrm{\mid }{B}_{jk}\mathrm{\mid }\mathrm{\parallel }{w}_j{\mathrm{\parallel }}_{\mathrm{\infty }},$ which are finite explicit bounds for the defining suprema.

Any finite-order derivative of the continuum series multiplies each coefficient by finitely many components of ${\kappa }_k$. The coefficient already has one factor $\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^{-1}$; after $r$ additional derivatives, its magnitude is bounded by a constant times $(1+\mathrm{\mid }k\mathrm{\mid }{)}^r{\prod }_a{\alpha }_{\epsilon }^{2\mathrm{\mid }{k}_a\mathrm{\mid }}$. These bounds are summable because each one-dimensional geometric series with a polynomial factor converges. Thus every differentiated series converges uniformly, and all derivatives are bounded. In particular, for a first derivative the factor $\mathrm{\mid }{\kappa }_k\mathrm{\mid }$ cancels the denominator $\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2$, so its norm is bounded by the displayed sum times $\mathrm{\mid }cQ\mathrm{\mid }\mathrm{/}{L}^d$. Periodicity permits the same bounds on Euclidean lifts.

First change $z=(x,v)$ while holding $z^{\mathrm{\prime }}=(y,w)$ fixed. The output difference has components $(v-\tilde{v},cQ({\mathcal{E}}_{\epsilon ,h}(x,y)-{\mathcal{E}}_{\epsilon ,h}(\tilde{x},y)))$. Its squared Euclidean norm is at most $\mathrm{\mid }v-\tilde{v}{\mathrm{\mid }}^2+(M_x^{\epsilon ,h}{)}^2\mathrm{\mid }x-\tilde{x}{\mathrm{\mid }}^2$, hence its norm is at most $\max \{1,M_x^{\epsilon ,h}\}\mathrm{\mid }z-\tilde{z}\mathrm{\mid }$. Next change $z^{\mathrm{\prime }}$ while holding $\tilde{z}$ fixed; this changes only the field component, by at most $M_y^{\epsilon ,h}\mathrm{\mid }{z}^{\mathrm{\prime }}-{\tilde{z}}^{\mathrm{\prime }}\mathrm{\mid }$. Adding these bounds gives the stated two-variable estimate with $L_{\epsilon ,h}$. The same block-norm calculation using the continuum derivative bounds from step 1.2 gives the estimate with $L_{\epsilon }$.