Weighted control on the negative half-space

Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Self-checked by GPT-6.1 Sol (OpenAI). Public domain (CC0).

The analytic-branch estimate controls a supremum of one-dimensional restriction norms. We now turn it into a norm estimate on the whole negative half-space, with an arbitrary moderate weight in all frequency variables. The two changes have separate mechanisms: a spatial cutoff brings complex frequencies back to the real axis, and a compact modulated convolution window localizes the weight. The final argument uses the exact disintegration already proved.

Read Weighted estimates from analytic root branches, Disintegrating half-space restriction norms, Rescaled symbols and stable strength. Fourier transforms, finite spectra and convex separation supplies Schwartz Fourier inversion; Metric and topological foundations supplies finite coordinates, scalar calculus and compact cutoffs; Banach estimates, quotient spaces and compact parameter arguments supplies the norm, extension and integration estimates.

Basic references are Grubb's Distributions and Operators (author's lecture notes), Melrose's Differential Analysis and Hörmander's The Analysis of Linear Partial Differential Operators. The proofs use the linked prerequisite lessons and any stated planned theorem.

The statement and the normalized root hypothesis

Write \((x,t)\in\mathbb R^d\times\mathbb R\), \(d\ge1\), and let \(P(z,s)\ne0\) be a polynomial. Assume that for some \(A\ge1\), every analytic root \(\tau\) on a complex ball of radius \(A\) with real center satisfies \[ \begin{gathered} P(z,\tau(z))=0 \\ \quad\Longrightarrow\\ \quad \sup\operatorname{Im}\tau\ge A+1 \text{ on that ball}. \end{gathered} \tag{1} \] Let \(S_P\) denote the exact full polynomial derivative norm. For a positive moderate joint weight \(k\), use the full-space and negative-half-space norms(equation 1 in Disintegrating half-space restriction norms)–(equation 2 in Disintegrating half-space restriction norms).

Theorem. There is \(\kappa\ge0\), depending only on \(P,A,d\), such that for every \(1\le p\le\infty\) and every such weight \(k\), there is \(C_{p,k}<\infty\) with \[ \|v\|^-_{p,k}\le C_{p,k}e^{\kappa L} \|P(D)v\|^-_{p,k/S_P}, \tag{2} \] whenever \(v\in C_c^\infty(\mathbb R^{d+1})\), \(L\ge0\), and \(|x|\le L\) on \(\operatorname{supp}v\cap\{t<0\}\). The exponent \(\kappa\) is independent of \(p,k\). Reducible polynomials and repeated factors are included.

Integrating actual Schwartz representatives

We first record the integral inequality needed for a quotient norm. Fix a moderate time weight \(\rho\), and let \(H(\eta,t)\) be a continuous Schwartz-valued function of the real parameter \(\eta\in\mathbb R^d\). Assume every Schwartz seminorm of \(H(\eta,\cdot)\) has an integrable parameter bound. Its actual pointwise integral and every fixed time derivative then define a time-Schwartz function \(h=\int H(\eta,\cdot)\,d\eta\), by dominated differentiation and the seminorm bounds. The fixed restriction norm satisfies \[ \left\|\int H(\eta,\cdot)\,d\eta\right\|^-_{p,\rho} \le\int\|H(\eta,\cdot)\|^-_{p,\rho}\,d\eta. \tag{3} \] For completeness, truncated Riemann sums converge to this integral in each Schwartz seminorm. On a compact parameter box, continuity gives uniform continuity in each of finitely many specified seminorms; choose a fine mesh for them. Integrable seminorm tails give convergence as the boxes increase. A diagonal choice controls successively every seminorm. The quotient map is continuous on Schwartz space, so its composition with each continuous unit-norm functional commutes with this limit and with the scalar integral. Scalar absolute-value bounds and the norm test B6 prove(3). Equivalently its norm is bounded by the integral of its norms at the finite-sum stage and in the limit. A completed quotient or an unproved Banach-valued integral is unnecessary.

All applications below have a joint Schwartz time family multiplied by a rapidly decreasing parameter kernel, so these integrable seminorm bounds hold.

From complex frequencies to a real integral

Assume first that \(P\) is irreducible and that \(k=k(s)\) depends only on the time frequency. Set \(g=P(D)v\) and \[ F(\eta)=\|\mathcal F_xg(\eta,\cdot)\|^-_{p,k/S_P(\eta,\cdot)}. \tag{4} \] This is continuous by the Disintegrating half-space restriction norms varying-weight argument and is integrable. Indeed the real partial transform of \(g\) decays with every time-Schwartz seminorm as \(|\eta|\to\infty\), \(k(s)\) has polynomial growth, and \(S_P(\eta,s)\) has a uniform positive lower bound from one constant highest derivative.

Choose \(\chi\in C_c^\infty(\mathbb R^d)\) equal to one on the unit ball and supported in the closed ball of radius two. Put \(B=\max(1,L)\), \(\chi_B(x)=\chi(x/B)\). Locality of \(P(D)\) implies that \(g\)'s negative-time spatial support is also inside the radius-\(L\) ball. Therefore \(\chi_Bg=g\) for \(t<0\).

The product-transform formula gives an actual full-time Schwartz representative \[ \begin{gathered} \mathcal F_x(\chi_Bg)(z,t) \\ =(2\pi)^{-d}\int_{\mathbb R^d} \\ \widehat\chi_B(z-\eta)\mathcal F_xg(\eta,t)\,d\eta, \\z\in\mathbb C^d. \end{gathered} \tag{5} \] To justify complex \(z\), insert the real-frequency inverse transform of \(g\); compact support of \(\chi_B\), the real Schwartz decay of that transform and a finite bound on \(e^{x\cdot\operatorname{Im}z}\) make Fubini absolutely convergent. Each time derivative obeys the same bounds. This representative has the same negative-time restriction as \(\mathcal F_xg(z,\cdot)\).

Let \(m=\deg P\). The finite jet-matrix shift, valid with complex base points, gives \(S_P(\eta,s)/S_P(z,s)\le(1+C_P|z-\eta|)^m\). Use the fixed time weight \(k/S_P(z,\cdot)\) in(3) and then this pointwise weight comparison. The result is \[ \begin{gathered} \|\mathcal F_xg(z,\cdot)\|^-_{p,k/S_P(z,\cdot)} \\ \le (2\pi)^{-d}\int \\ |\widehat\chi_B(z-\eta)|\\ (1+C_P|z-\eta|)^m F(\eta)\,d\eta. \end{gathered} \tag{6} \] No parameter-dependent norm is moved through an analytic argument here: for each fixed \(z\),(3) uses one fixed weight.

For every integer \(J\), the compact spatial transform and integration by parts in a coordinate of largest complex modulus give \[ \begin{gathered} |\widehat\chi_B(w)| \\ \le C_J B^d e^{2B|\operatorname{Im}w|} (1+B|w|)^{-J}. \end{gathered} \tag{7} \] The factor \(B^d\) is its exact scaling factor. For \(|Bw|\le1\), use the compact integral bound; otherwise integrate \(J\) derivatives of \(\chi\), as in the Analytic norms and propagation on complex balls full-complex decay proof. No real-frequency-only decay estimate is substituted for this bound.

Let \(R\) be the radius in the Weighted estimates from analytic root branches strip theorem, and take \(J\ge m\). Since \(B\ge1\), the kernel in(6) is uniformly bounded for \(|\operatorname{Im}w|<R\) by \[ C B^d e^{2RB}\le C' e^{(2R+1)B}. \tag{8} \] The last inequality absorbs the fixed polynomial \(B^d\) into \(e^B\). The constants depend on the symbol and cutoff, not on the time weight or norm exponent. Taking the strip supremum and then the Weighted estimates from analytic root branches estimate yields \[ \begin{gathered} \sup_{\xi\in\mathbb R^d} \|\mathcal F_xv(\xi,\cdot)\|^-_{p,k} \\ \le C_0 e^{\kappa_0 L}\int_{\mathbb R^d}F(\eta)\,d\eta, \\\kappa_0=\kappa_{\rm WE}+2R+1. \end{gathered} \tag{9} \] Here \(B\le L+1\), so its extra fixed exponential is absorbed into \(C_0\). Most importantly, \(C_0,\kappa_0\) are uniform over all time-only weights and all \(p\): the Weighted estimates from analytic root branches estimate is uniform in them, the integral norm inequality has coefficient one and every comparison above is a pointwise symbol comparison.

A modulated window allows an arbitrary joint weight

Now let \(k=k(\eta,s)\) be a joint moderate weight with constants \(C_k,N_k\). Choose a nonnegative \(\phi\in C_c^\infty(\mathbb R^d)\), supported in the unit ball, with \(\int\phi=1\). Thus \(\widehat\phi(0)=1\). For each real \(\eta\), set \[ \begin{gathered} \phi_\eta(x)=e^{ix\cdot\eta}\phi(x),\\ \qquad v_\eta=\phi_\eta*_xv,\\ \qquad \mathcal F_xv_\eta(\xi,t) =\widehat\phi(\xi-\eta)\mathcal F_xv(\xi,t). \end{gathered} \tag{10} \] Tangential convolution commutes with \(P(D)\), so \(P(D)v_\eta=\phi_\eta*_xg\). All these functions are smooth and compactly supported; at negative times their spatial support lies inside the radius-\((L+1)\) ball.

Apply(9) to \(v_\eta\) with time-only weight \(k_\eta(s)=k(\eta,s)\). Its constants are uniform in \(\eta\), since they are uniform over time weights. Evaluate its left side at \(\xi=\eta\), where the window transform equals one. Moderation of the joint weight gives \[ \begin{gathered} V(\eta):\\ =\|\mathcal F_xv(\eta,\cdot)\|^-_{p,k_\eta} \\ \le C_1 e^{\kappa_0 L}\int_{\mathbb R^d} K(\eta-\xi)G(\xi)\,d\xi, \end{gathered} \tag{11} \] where \(G(\xi)=\|\mathcal F_xg(\xi,\cdot)\|^-_{p,k_\xi/S_P(\xi,\cdot)}\) and \(K(h)=|\widehat\phi(-h)|(1+C_k|h|)^{N_k}\). The scalar window factor comes out of the restriction norm by homogeneity. The weight comparison is uniform in \(s\) and retains the same \(S_P(\xi,s)\) on both sides. The factor \(e^{\kappa_0}\) from the window support is absorbed into \(C_1\).

The real Schwartz transform of \(\phi\) makes \(K\in L^1\), for any fixed moderation exponent. Young's \(L^1*L^p\) inequality and the exact Disintegrating half-space restriction norms identity now give \[ \begin{aligned} \|v\|^-_{p,k} &=(2\pi)^{-d/p}\|V\|_p\\ &\le C_1 e^{\kappa_0 L}\|K\|_1 (2\pi)^{-d/p}\|G\|_p\\ &=C_1\|K\|_1 e^{\kappa_0 L}\|P(D)v\|^-_{p,k/S_P}. \end{aligned} \tag{12} \] This is valid at \(p=1,\infty\) as well as every intermediate exponent. The dependence on the weight is confined to the finite constant \(\|K\|_1\); it does not change \(\kappa_0\).

Reducible polynomials and multiplicities

Factor \(P=cP_1\cdots P_\ell\), with \(c\ne0\) and each \(P_j\) irreducible, repeating factors with their multiplicities. Every analytic root of a factor is an analytic root of \(P\), so(1) passes to each factor with the same \(A\). The already written product derivative-norm lemma gives constants \(c_*,C_*>0\) such that \[ \begin{gathered} c_*\prod_j S_{P_j}\le S_{P_1\cdots P_\ell} \le C_*\prod_jS_{P_j},\\ \qquad S_P=|c|S_{P_1\cdots P_\ell}. \end{gathered} \tag{13} \] Apply the irreducible estimate successively to \(v,P_1(D)v,\ldots,P_{\ell-1}(D)\cdots P_1(D)v\), with weights \(k,k/S_{P_1},\ldots,k/\prod_{j<\ell}S_{P_j}\). These are positive moderate weights. The intermediate differential operators preserve compact support and the negative-time spatial bound by locality. Their individual exponential coefficients depend only on their symbols, so their sum is independent of all the weights.

The final restriction norm uses \(k/\prod_jS_{P_j}\). By(13) it is bounded by a constant times the norm with weight \(k/S_{P_1\cdots P_\ell}\). Scalar norm homogeneity converts the resulting last term exactly to \(\|P(D)v\|^-_{p,k/S_P}\); the scalar \(c\) cancels its identical norm factor. This proves(2) for all nonzero \(P\). A nonzero constant has \(S_P=|P|\) and gives equality. Repetition introduces another valid factor estimate, not a distinct-root assumption.

In tangential dimension zero, the full norm is the one-dimensional norm. The Weighted estimates from analytic root branches zero-dimensional argument proves the same estimate directly, with no cutoff or window. Thus that dimension is included with its stated convention.

Exercises with complete solutions

Exercise 1 — basic: the window normalization. Explain why \(\phi(0)=1\) cannot replace \(\widehat\phi(0)=1\) in(10). Construct a smooth compact example with value one at zero and transform zero at zero.

Solution. Evaluation at the selected frequency \(\xi=\eta\) produces the multiplier \(\widehat\phi(0)=\int\phi\), not the physical-space value. Choose a smooth cutoff \(b\) supported in a small ball about zero with \(b(0)=1\) and positive integral. Choose a nonnegative smooth bump \(a\) in another small ball inside the unit ball, disjoint from zero, with positive integral. The function \[ \begin{gathered} \phi=b-\frac{\int b}{\int a}a \\ \quad\text{satisfies}\\ \quad \phi(0)=1,\\ \qquad \widehat\phi(0)=0. \end{gathered} \tag{14} \] At the selected frequency it would erase the very fiber being estimated. A valid window is obtained by dividing any nonnegative nonzero compact bump by its positive integral. Its modulation then shifts the transform by exactly \(\eta\), and tangential convolution has the product-transform identity with no extra Fourier factor.

Exercise 2 — intermediate: constants versus exponential rate. Show that the spatial scaling factor \(B^d\), \(B\ge1\), can be absorbed in \(e^B\) with a degree-dependent constant. Explain why a larger moderation exponent for the joint weight need not increase \(\kappa\).

Solution. For \(d\ge1\), differentiation gives \[ \begin{gathered} \frac{d}{dB}(B^de^{-B}) =B^{d-1}e^{-B}(d-B),\\ \qquad B^d\le d^de^{-d}e^B\\ \quad(B\ge1). \end{gathered} \tag{15} \] Its maximum on this interval occurs at \(B=d\), including the endpoint \(d=1\). Dimension zero permits the weaker constant one directly. Thus the one extra support exponential in(8) suffices regardless of the scaling polynomial.

The joint weight enters later through \(K(h)=|\widehat\phi(-h)|(1+C_k|h|)^{N_k}\). For every fixed \(N_k\), choose a Schwartz decay bound of order greater than \(N_k+d\). This proves a finite \(L^1\) norm for \(K\). Increasing \(N_k\) may increase that finite constant, but the window still has spatial radius one and the first-stage exponent \(\kappa_0\) was uniform over time weights. Therefore the same \(\kappa\) remains valid.

Exercise 3 — advanced: repeated factors and the exact derivative norm. For the one-variable symbol \(P(s)=(s-2i)^2\), calculate \(S_P(s)\) on the real axis and compare it to the product of the two first-order derivative norms. Explain its role in the repeated-factor step.

Solution. Put \(R(s)=s-2i\). For real \(s\), \(S_R(s)=\sqrt{s^2+5}\). The squared norm of \(P\) includes its value, first derivative \(2(s-2i)\) and second derivative \(2\): \[ \begin{gathered} S_P(s)^2\\ =(s^2+4)^2+4(s^2+4)+4\\ =(s^2+6)^2,\\S_R(s)^2=s^2+5. \end{gathered} \tag{16} \] Consequently \(S_P(s)=s^2+6\), while the product of the first-order norms is \(s^2+5\). Their ratio lies between one and \(6/5\). The product weight is comparable to the exact derivative norm and is not literally identical to it. Applying the first-order estimate twice gives the weight \(k/(s^2+5)\); this comparison converts it to the target weight \(k/(s^2+6)\) with a finite constant. Its constant root \(2i\) satisfies(1) with \(A=1\), and the one-variable half-line inverse gives the resulting bound with no spatial exponential.

References