ON CERTAIN QUOTIENTS OF HARDY SPACES

Svante Janson

For any real number η, let Ω_{η} be the half–plane {z : Im z > η} in the complex plane and let, for 1 ≤ p ≤ ∞, H^{p}(Ω_{η}) denote the usual Hardy space

(1)  H^{p}(Ω_{η}) = {F ∈ H(Ω_{η}) : ||F ||_{H^{p}(Ω_{η})} = sup_{y>η} ||F_{y}||_{L^{p}(R)} < ∞},

where H(Ω) denotes the space of holomorphic functions in Ω and

(2)  F_{y}(x) = F (x + iy).

It is well-known that the spaces H^{p}(Ω_{η}) are Banach spaces and that H^{∞}(Ω_{η}) is a commutative Banach algebra. The translation S_{δ} : f ↦ f(z − iδ) is an isometry H^{p}(Ω_{η}) → H^{p}(Ω_{η+δ}), while the identity (or rather restriction) yields a continuous (but not isometric) inclusion H^{p}(Ω_{η_{1}}) ⊂ H^{p}(Ω_{η_{2}}) whenever η_{1} < η_{2}.

Let Π^{p} = ⋃_{−∞<η<∞} H^{p}(Ω_{η}), thus Π^{p} is the linear space of all functions which are holomorphic in some Ω_{η} and belong to H^{p} there, with two functions identified if they coincide on some Ω_{η}.

The space e^{iz}H^{p}(Ω_{η}) = {e^{iz}F : F ∈ H^{p}(Ω_{η})} is a closed subspace of H^{p}(Ω_{η}) (and an ideal if p = ∞). If η_{1} < η_{2}, then the inclusion map H^{p}(Ω_{η_{1}}) → H^{p}(Ω_{η_{2}}) induces a map j : H^{p}(Ω_{η_{1}})/e^{iz}H^{p}(Ω_{η_{1}}) → H^{p}(Ω_{η_{2}})/e^{iz}H^{p}(Ω_{η_{2}}).

It turns out that j is a bijection for all η_{1} < η_{2}; hence j is an isomorphism

(3)  H^{p}(Ω_{η_{1}})/e^{iz}H^{p}(Ω_{η_{1}}) ≅ H^{p}(Ω_{η_{2}})/e^{iz}H^{p}(Ω_{η_{2}})

and it follows that the inclusion H^{p}(Ω_{η}) → Π^{p} induces an (algebraic) isomorphism

(4)  H^{p}(Ω_{η})/e^{iz}H^{p}(Ω_{η}) ≅ Π^{p}/e^{iz}Π^{p},  −∞ < η < ∞.

This result was proved by Frankfurt and Rovnyak (1975, 1978) in their study of finite convolution operators. It is easily seen that H^{p}(Ω_{η_{1}}) ∩ e^{iz}H(Ω_{η_{2}}) = e^{iz}H^{p}(Ω_{η_{1}}), which implies that j is an injection, so the main part of the assertion is that j is surjective, or equivalently that every F ∈ H^{p}(Ω_{η_{2}}) equals some G ∈ H^{p}(Ω_{η_{1}}) modulo e^{iz}H^{p}(Ω_{η_{2}}). The purpose of this note is to give a simple constructive proof of this assertion. For notational convenience we assume η_{1} = 0, which implies the general case by translations.

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Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine

Theorem. Let η > 0. Assume that F ∈ H^{p}(Ω_{η}) for some p, 1 ≤ p ≤ ∞, and define

(5)  T_{η}F (z) = \frac{1}{2πi} ∫_{Im ζ=−η} (\frac{1}{ζ} − \frac{1}{ζ + 2iη} ) e^{iζ}F (z − ζ) dζ,  z ∈ Ω_{0}.

Then T_{η}F ∈ H^{p}(Ω_{0}) and T_{η}F − F ∈ e^{iz}H^{p}(Ω_{η}). Moreover, T_{η} is a bounded linear map H^{p}(Ω_{η}) → H^{p}(Ω_{0}), and it induces a map H^{p}(Ω_{η})/e^{iz}H^{p}(Ω_{η}) → H^{p}(Ω_{0})/e^{iz}H^{p}(Ω_{0}) which is the inverse of the map H^{p}(Ω_{0})/e^{iz}H^{p}(Ω_{0}) → H^{p}(Ω_{η})/e^{iz}H^{p}(Ω_{η}) induced by the inclusion H^{p}(Ω_{0}) → H^{p}(Ω_{η}).

Proof. Let

K(ζ) = \frac{1}{2πi}(\frac{1}{ζ} − \frac{1}{ζ + 2iη} )e^{iζ} = \frac{η}{πζ(ζ + 2iη)}e^{iζ}.

Then

||K_{η}||_{L^{1}} = ∫_{−∞}^{∞} |K(x − iη)| dx = ∫_{−∞}^{∞} \frac{η}{π|x − iη||x + iη|}e^{η} dx

= \frac{e^{η}}{π} ∫_{−∞}^{∞} \frac{η}{x^{2} + η^{2}} dx = e^{η} < ∞.

Since H^{p}(Ω_{η}) ⊂ H^{∞}(Ω_{η′}) for every η′ > η, it follows that the integral in (5) converges for every z ∈ Ω_{0} and T_{η}F (z) is easily seen to be holomorphic. Moreover, (T_{η}F)_{y} = K_{−η} ∗F_{y+η} and thus by Young’s inequality,

||(T_{n}F)_{y}||_{L^{p}} ≤ ||K_{−η}||_{L^{1}}||F_{y+η}||_{L^{p}} ≤ e^{η}||F ||_{H^{p}(Ω_{η})}, y > 0.

Thus T_{η}F ∈ H^{p}(Ω_{0}).

The kernel K(ζ) has simple poles at ζ = 0 and ζ = −2iη and the residue at 0 is 1/2πi. Hence, if δ > 0 and y = Im z > η + δ, Cauchy’s integral formula (with simple estimates for large |ζ|) yields

T_{η}F (z) = ∫_{Im ζ=−η} K(ζ)F (z − ζ) dζ = ∫_{Im ζ=δ} K(ζ)F (z − ζ) dζ + F (z).

Assume for simplicity y > 2η and let δ = y − 2η. Then

T_{η}F (z) − F (z) = ∫_{Im ζ=y−2η} K(ζ)F (z − ζ) dζ = − ∫_{Im ζ=2η} K(z − ζ)F (ζ) dζ
= e^{iz}G(z),

with

G(z) = − ∫_{Im ζ=2η} e^{−iz}K(z − ζ)F (ζ) dζ = − ∫_{Im ζ=2η} \frac{η}{π(z − ζ)(z − ζ + 2iη)}e^{−iζ}F (ζ) dζ.

Young’s inequality yields

||G_{y}||_{L^{p}} ≤ ∫_{−∞}^{∞} \frac{η}{π|t + i(y − 2η)||t + iy|} dt ||e^{2η}F_{2η}||_{L^{p}}

≤ ∫ \frac{η}{π(t^{2} + (y − 2η)^{2})} dt e^{2η}||F_{2η}||_{L^{p}} = \frac{η}{y − 2η} e^{2η}||F_{2η}||_{L^{p}}, y > 2η.

Hence G ∈ H^{p}(Ω_{3η}) and

T_{η}F (z) − F (z) = e^{iz}G(z) ∈ H^{p}(Ω_{η}) ∩ e^{iz}H^{p}(Ω_{3η}) = e^{iz}H^{p}(Ω_{η}).

The remaining assertions follow easily. □

The theorem can also be derived by Fourier analysis (this is in fact how the formula (5) was found). We sketch the argument which is based on well-known extensions of the Paley–Wiener theorem to p ≠ 2. For p > 2, the Fourier transforms are interpreted as distributions on R. Cf. for example Garnett (1981), Chapter II (in particular Exercise 2) for some background.

If F ∈ H^{p}(Ω_{η}), then F̂_{y}(ξ) = 0 for ξ < 0, y > η and F̂_{y_{2}}(ξ) = e^{−(y_{2}−y_{1})ξ}F̂_{y_{1}}(ξ) for y_{2} > y_{1} > η. Hence F̂(ξ) = e^{yξ}F̂_{y}(ξ), y > η, is well-defined. The mapping F ↦ F̂ is an isomorphism of H^{p}(Ω_{0}) onto Ĥ^{p} = {f ∈ L̂^{p}(R) : supp f ⊂ [0, ∞)} and, more generally, of H^{p}(Ω_{η}) onto e^{ηξ}Ĥ^{p} = {e^{ηξ}f(ξ) : f ∈ Ĥ^{p}}, −∞ < η < ∞.

If F ∈ H^{p}(Ω_{η}), then ê^{iz}F (ξ) = F̂(ξ − 1), and thus F ↦ F̂ maps e^{iz}H^{p}(Ω_{η}) onto {e^{ηξ}f(ξ − 1) : f ∈ Ĥ^{p}} = {e^{ηξ}f(ξ) : f ∈ L̂^{p}, supp f ⊂ [1, ∞)}. Hence e^{iz}H^{p}(Ω_{η}) = {f ∈ H^{p}(Ω_{η}) : supp f̂ ⊂ [1, ∞)}.

If η > 0 and F ∈ H^{p}(Ω_{η}), then F̂(ξ) = e^{ηξ}ĝ(ξ) for some ĝ ∈ Ĥ^{p}. In general, F̂ is, of course, not an element of Ĥ^{p}, but if we modify F̂ on [1, ∞), for example to Ĝ(ξ) = e^{η min(ξ,2−ξ)}ĝ(ξ) = e^{η−η|ξ−1|}ĝ(ξ), then supp(F̂−Ĝ) ⊂ [1, ∞) and Ĝ ∈ Ĥ^{p} because G = k∗g where k(x) = e^{η}e^{ix}P_{η}(x) ∈ L^{1} (P_{η} is the Poisson kernel) since k̂(ξ) = e^{η}P̂_{η}(ξ − 1) = e^{η−η|ξ−1|}. Thus G ∈ H^{p}(Ω_{0}) and F − G ∈ e^{iz}H^{p}(Ω_{η}) as required.

In fact, it is easily seen that k(x) = K(x − iη) and G = T_{η}F as given in (5). We also obtain Ĝ(ξ) = e^{η−η|ξ−1|}ĝ(ξ) = e^{η−η|ξ−1|−ηξ}F̂(ξ), or

(6)  T̂_{η}F (ξ) = m_{η}(ξ)F̂(ξ),

with

(7)  m_{η}(ξ) = \{ \begin{matrix}1 & ξ ≤ 1 \\ e^{−2η(ξ−1)} & ξ ≥ 1. \end{matrix}

Remark 1. Since m_{η_{1}}(ξ)m_{η_{2}}(ξ) = m_{η_{1}+η_{2}}(ξ) we obtain the semigroup property T_{η_{1}}T_{η_{2}} = T_{η_{1}+η_{2}}.

Remark 2. The proof above shows that ||T_{η}||_{H^{p}(Ω_{η}),H^{p}(Ω_{0})} ≤ e^{η}. In fact, considering F such that F̂ ∈ C_{0}^{∞} (say) with supp F̂ ⊂ [1 − ε, 1 − ε + δ], where 0 < δ < ε < 1, it is easily seen that

(8)  ||T_{η}||_{H^{p}(Ω_{η}),H^{p}(Ω_{0})} = e^{η},  1 ≤ p ≤ ∞.

This can easily be extended to show that any (not necessarily linear) operator T : H^{p}(Ω_{η}) → H^{p}(Ω_{0}) such that T F − F ∈ e^{iz}H^{p}(Ω_{η}) for every F, has norm at least e^{η}. Hence T_{η} has minimal norm among all such operators.

Remark 3. If F ∈ Π^{p}, then T_{η}F is defined for every η that is large enough. These functions are in general different elements of H^{p} = H^{p}(Ω_{0}), although they all represent the same element of H^{p}/e^{iz}H^{p}. In order to study what happens when η → ∞, let us for simplicity assume F ∈ H^{p}. We identify a function F ∈ H^{p} = H^{p}(Ω_{0}) with its boundary values in L^{p} = L^{p}(R).

If 1 < p < ∞, then it is an immediate consequence of Riesz’ theorem on boundedness of harmonic conjugation that the operators I_{a} defined for −∞ < a < ∞ by (I_{a}f)^{∧} = χ_{(−∞,a)}f̂ are bounded operators on L^{p}, with norm independent of a. Moreover, a ↦ I_{a}f is a continuous map into L^{p} for every f ∈ L^{p}. It follows from (6) and (7) that

T_{η}F = ∫_{1}^{∞} 2ηe^{−2η(a−1)}I_{a}F da.

Hence ||T_{η}||_{H^{p}} ≤ ||I_{0}||_{L^{p}} = C_{p}, uniformly in η, and T_{η}F → I_{1}F in H^{p} as η → ∞ for every F ∈ H^{p} (and thus for every F ∈ Π^{p}).

On the other hand, if p = 1 or p = ∞, then I_{a} is not bounded on L^{p} and I_{1} is not bounded on H^{p}. It follows easily that there exists F ∈ H^{p} such that ||T_{η}F ||_{H^{p}} is unbounded as η → ∞, and ||T_{η}||_{H^{p}} → ∞ as η → ∞.

Acknowledgment. I am grateful to James Rovnyak for helpful discussions.

References

R. Frankfurt and J. Rovnyak (1975), Finite convolution operators, J. Math. Anal. Appl. 49, 347–374. R. Frankfurt and J. Rovnyak (1978), Recent results and unsolved problems on finite convolution operators,

Linear spaces and approximation, Proceedings, Oberwolfach 1977, Birkhäuser, Basel, pp. 133–150. J. Garnett (1981), Bounded analytic functions, Academic Press, New York.

Svante Janson, Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden

   E-mail address: svante.janson@math.uu.se