Carleson real line to torus transfer

Let $P(x)={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le m}{a}_k{e}^{2\pi ikx}$ be a trigonometric polynomial. Its Fourier coefficients are $a_k$: the integral of $e^{2\pi i(k-j)t}$ on $[0,1]$ is one for $k=j$ and zero otherwise, by direct integration for the nonzero integer $k-j$. For $0<\epsilon <1$ put $u_{\epsilon }(x)=\varphi (\epsilon x)P(x)$, a Schwartz function: the derivatives of $P$ are bounded and the scaled $\varphi$ and all its derivatives decay to every order, so the Leibniz formula proves every Schwartz seminorm finite. Direct substitution in the absolutely convergent Fourier integral gives $\widehat{u_{\epsilon }}(\xi )={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le m}{a}_k{\epsilon }^{-1}\psi ((\xi -k)\mathrm{/}\epsilon )\mathrm{.}$ The $k$th summand is supported in $[k-\epsilon \mathrm{/}8,k+\epsilon \mathrm{/}8]$. Thus for every integer $N\ge 0$ the interval $[-N-1\mathrm{/}2,N+1\mathrm{/}2]$ contains exactly the entire summands with $\mathrm{\mid }k\mathrm{\mid }\le N$ and misses the other ones. F4 then gives the exact identity $T_{N+1\mathrm{/}2}{u}_{\epsilon }(x)-T_{-N-1\mathrm{/}2}{u}_{\epsilon }(x)=\varphi (\epsilon x)S_{N}P(x)\mathrm{.}$ The half-integer cutoffs avoid every endpoint frequency; no half-weight term occurs.

We prove the precise periodic averaging limit used below. Let $w$ be continuous, nonnegative and bounded by $C(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-2}$, and let $G$ be continuous and one-periodic. Then $\epsilon {\int }_{\mathbb{R}}w(\epsilon x)G(x)dx⟶\left({\int }_{\mathbb{R}}w\right){\int }_0^{1}G(t)dt\mathrm{.}$ Decomposing the line into $n+[0,1)$ and substituting gives the left side as ${\int }_0^{1}G(t)R_{\epsilon }(t)dt$, where $R_{\epsilon }(t)=\epsilon {\sum }_{n\in \mathbb{Z}}w(\epsilon (n+t))$. This rearrangement is absolute: $\mathrm{\mid }G\mathrm{\mid }$ is bounded, $\int w<\mathrm{\infty }$, and F8 followed by F9 applies. These Riemann sums converge uniformly for $t$ in $[0,1]$ to $\int w$. To verify uniformity, partition the line into cells $[\epsilon (n+t),\epsilon (n+t+1))$. For cells meeting $[-R,R]$, the difference between the left-endpoint sum and the integral is at most $(2R+2){\omega }_R(\epsilon )$, where ${\omega }_R$ is the modulus of continuity of $w$ on $[-R-1,R+1]$, and $0<\epsilon \le 1$. For the remaining cells the sum and integral of the decay majorant are at most $C\mathrm{/}(1+R)$, uniformly in $t$ and $\epsilon$, by comparison of the monotone tails of $(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-2}$ with their integrals. First choose $R$ large and then $\epsilon$ small. The resulting uniform convergence allows integration against bounded $G$ and proves the displayed limit.

For every finite cutoff bound $J$, the identity in step 1.1 implies $\mathrm{\mid }\varphi (\epsilon x)\mathrm{\mid }{\max }_{0\le N\le J}\mathrm{\mid }{S}_{N}P(x)\mathrm{\mid }\le 2C_{\mathbb{R}}{u}_{\epsilon }(x)\mathrm{.}$ Raise to $p$, integrate and use the assumed real-line bound to obtain $\epsilon {\int }_{\mathbb{R}}\mathrm{\mid }\varphi (\epsilon x){\mathrm{\mid }}^p({\max }_{0\le N\le J}\mathrm{\mid }{S}_{N}P(x)\mathrm{\mid }{)}^{p}dx\le (2D_p{)}^p\epsilon {\int }_{\mathbb{R}}\mathrm{\mid }\varphi (\epsilon x){\mathrm{\mid }}^p\mathrm{\mid }P(x){\mathrm{\mid }}^{p}dx\mathrm{.}$ Both periodic factors in this display are bounded and continuous.

Apply step 1.2 with $w=\mathrm{\mid }\varphi {\mathrm{\mid }}^p$. Its hypotheses hold by Schwartz decay and continuity, choosing a decay exponent $M$ with $Mp\ge 2$. Also $0<\int \mathrm{\mid }\varphi {\mathrm{\mid }}^p<\mathrm{\infty }$, since $\varphi$ is a nonzero continuous Schwartz function. Taking $\epsilon$ to zero in step 2.1 and dividing by this positive window integral yields $\parallel {\max }_{0\le N\le J}\mathrm{\mid }{S}_{N}P\mathrm{\mid }{\parallel }_{L^p([0,1])}\le 2D_p\mathrm{\parallel }P{\mathrm{\parallel }}_{L^p([0,1])}\mathrm{.}$ This has normalized Haar measure exactly: the periodic averaging limit contains ${\int }_0^1$, with no interval-length factor. The constant is independent of $J$ and the degree of $P$.

For any $f\in L^p(\mathbb{T})$, normalized measure and F7 give $\mathrm{\parallel }f{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_p$, so its coefficients and partial sums in F3 are defined. By F5 and F6, the polynomials $P_j={\sigma }_{j}f$ converge to $f$ in $L^p$. For each fixed $J$, ${\max }_{0\le N\le J}\mathrm{\parallel }{S}_N(P_j-f){\mathrm{\parallel }}_{\mathrm{\infty }}\le (2J+1)\mathrm{\parallel }{P}_j-f{\mathrm{\parallel }}_1\le (2J+1)\mathrm{\parallel }{P}_j-f{\mathrm{\parallel }}_p⟶0,$ because every Fourier coefficient difference has magnitude at most its $L^1$ norm and every character has modulus one. Hence the finite maxima for $P_j$ converge uniformly to the finite maximum for $f$. Step 3.1 and Minkowski's norm continuity give the same bound $2D_p\mathrm{\parallel }f{\mathrm{\parallel }}_p$ for that finite maximum. Finally these nonnegative maxima increase as $J$ tends to infinity; F10 gives $\parallel {\sup }_{N\ge 0}\mathrm{\mid }{S}_{N}f\mathrm{\mid }{\parallel }_{L^p(\mathbb{T})}\le 2D_p\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathbb{T})}\mathrm{.}$ Their countable supremum is measurable. The zero input and zero polynomial cases follow directly, and $N=0$ was included in the exact cutoff identity. Apply this argument separately to each $p$ strictly between one and infinity. AC is inherited through F11; the approximants here are the explicitly specified Fejer means. This proves the conditional transfer without assuming that the real-line bound has already been established elsewhere.